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Integral de 2*sqrt(1-x^2-y^2)dy dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
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      |      2*\/  1 - x  - y   dy
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-\/  1 - x                        
$$\int\limits_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} 2 \sqrt{- y^{2} + \left(1 - x^{2}\right)}\, dy$$
Integral(2*sqrt(1 - x^2 - y^2), (y, -sqrt(1 - x^2), sqrt(1 - x^2)))
Respuesta (Indefinida) [src]
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 | 2*\/  1 - x  - y   dy = C + 2*|<                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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                                 \\                                                                                         \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /                                                                                                          /
$$\int 2 \sqrt{- y^{2} + \left(1 - x^{2}\right)}\, dy = C + 2 \left(\begin{cases} - \frac{2 i x^{2} y \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} - \frac{2 i x^{2} \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} + \frac{\pi x^{2} \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} - \frac{2 i y^{3} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} + \frac{2 i y \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} + \frac{2 i \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} - \frac{\pi \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} & \text{for}\: \left|{\frac{y^{2}}{x^{2} - 1}}\right| > 1 \\\frac{x^{2} y \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} + \frac{x^{2} \left(1 - x^{2}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} + \frac{y^{3} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} - \frac{y \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} - \frac{\left(1 - x^{2}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\right)$$
Respuesta [src]
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|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                     4*I*y *\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|   2*pi*   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|   4*I*y*\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|   4*I*y*x *\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|                                                                                                                                     2*pi*x *   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|   4*I*   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|   4*I*x *   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                             
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|      |      |                              ____________________                                                           ____________________                                                         ____________________                                                                                                                                                                                                                                                                      |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                 |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                 |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                    |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                                                                                                                                                      |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                                                  |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                                                                                                                                                                                                 |   ____________________|                                                                                                           |   ____________________|                                                                                              
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|      |      |                       4*I*\/  polar_lift\1 - x /                                                 12*I*y *\/  polar_lift\1 - x /                                                4*I*x *\/  polar_lift\1 - x /                                                                      4*I*\1 - x /                                                                                         4*I*x *\1 - x /                                                                             \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                 \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                 \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                    \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                           2*pi*y*\1 - x /                                                                                                            \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                                                  \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                                                         2*pi*y*x *\1 - x /                                                                                                                      \\/  polar_lift\1 - x / /                                                                                                           \\/  polar_lift\1 - x / /                                             |   y   |                                        
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|      |      |        /               y                2     /               y                     /               y                2     /               y                     /               y                2     /               y             |        /               y                2     /               y          |   /           /     2\    |        /               y                2     /               y          |   /           /     2\                                 |         /               2                      /               2         |                                                                       |         /               2                      /               2         |                                                                      |         /               2                      /               2         |                                                               |         /               2                      /               2         |                                       /               y           |        /               y                2     /               y          |           /     2\                                         |         /               2                      /               2         |                                                                                               |         /               2                      /               2         |                                                                                                             |         /               2                      /               2         |                                                              /               y           |        /               y                2     /               y          |           /     2\       /               y           |        /               y                2     /               y          |           /     2\       /               y           |        /               y                2     /               y          |           /     2\                                                       
|      |      |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*\/  polar_lift\1 - x /    |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*\/  polar_lift\1 - x /                                 |        /               y                2     /               y          |                                                                       |        /               y                2     /               y          |                                                                      |        /               y                2     /               y          |                                                               |        /               y                2     /               y          |                                      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /                                         |        /               y                2     /               y          |                                                                                               |        /               y                2     /               y          |                                                                                                             |        /               y                2     /               y          |                                                             /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /                                                       
|      |      |      /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\    |      /                   /     2\           /                   /     2\ |                           |      /                   /     2\           /                   /     2\ |                                                        |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                       |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                      |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                               |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                     /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                            |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                                               |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                                                             |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                            /                   /     2\  |      /                   /     2\           /                   /     2\ |                        /                   /     2\  |      /                   /     2\           /                   /     2\ |                        /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                                          
|      |      |    \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /    \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                           \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                        |      /                   /     2\           /                   /     2\ |                                                                       |      /                   /     2\           /                   /     2\ |                                                                      |      /                   /     2\           /                   /     2\ |                                                               |      /                   /     2\           /                   /     2\ |                                   \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                            |      /                   /     2\           /                   /     2\ |                                                                                               |      /                   /     2\           /                   /     2\ |                                                                                                             |      /                   /     2\           /                   /     2\ |                                                          \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                      \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                      \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                   ________       ________
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                       \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                      \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                               \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                                                                                                                           \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                               \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                             \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            /      2       /      2 
|      |      <                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          dy   for \/  1 - x   > -\/  1 - x  
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 ________________________                                                                                                                                                                                                                                                                                                                                                                                                                                         ________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          ____________________ /                                                                                 2                     \                                                                                                                ____________________ /                                                                                 2                     \          /              2                   /                                                                                 2                     \                                           ____________________ /                                                                                 2                     \                                                                                                                                            /              2                   /                                                                                 2                     \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         /           /     2\  |                         2*y                                                2*y*x                      |                                                                                                           3   /           /     2\  |                         2*y                                                2*y*x                      |         /              y           /     2\ |                         2*y                                                2*y*x                      |     /           y           \        2   /           /     2\  |                         2*y                                                2*y*x                      |                                                                                                                                     2     /              y           /     2\ |                         2*y                                                2*y*x                      |     /           y           \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   2*y*\/  polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------|                                                                                                        2*y *\/  polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------|   2*   /   1 - ------------------ *\1 - x /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|   2*y*x *\/  polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------|                                                                                                                                  2*x *   /   1 - ------------------ *\1 - x /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |        ________________________                            ________________________                   |                                                                                                                                     |        ________________________                            ________________________                   |       /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|                                  |        ________________________                            ________________________                   |                                                                                                                                         /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |       /              2                                    /              2                            |                                                                                                                                     |       /              2                                    /              2                            |     \/         polar_lift\1 - x /           |       /              2                                    /              2                            |     |  /           /     2\ |                                  |       /              2                                    /              2                            |                                                /     2\     /           y           \                                                 \/         polar_lift\1 - x /           |       /              2                                    /              2                            |     |  /           /     2\ |                                               2 /     2\     /           y           \                                                                                                                                                                                                                                                                                                                                                                                                           
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |      /              y                     /     2\       /              y                     /     2\|                                                                                                                                     |      /              y                     /     2\       /              y                     /     2\|                                             |      /              y                     /     2\       /              y                     /     2\|     \\/  polar_lift\1 - x / /                                  |      /              y                     /     2\       /              y                     /     2\|                                            2*y*\1 - x /*asin|-----------------------|                                                                                         |      /              y                     /     2\       /              y                     /     2\|     \\/  polar_lift\1 - x / /                                          2*y*x *\1 - x /*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                           
|      |      |                                                                                                                                                                                                                                                                                                                                         ____________________                                                                                                                                                              ____________________                                                       ____________________                                                     |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                                                                                                     |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                             |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                                |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                             |   ____________________|                                                                                         |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                                                            |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                           
|      |      |                                                                                                                                                                                                                                                                                                                                        /           /     2\                                                                          /     2\                                                                        2   /           /     2\                                                   2   /           /     2\                                                      |    /                  /     2\                         /                  /     2\                    |                                                2 /     2\                                                                           |    /                  /     2\                         /                  /     2\                    |                                             |    /                  /     2\                         /                  /     2\                    |                                                                |    /                  /     2\                         /                  /     2\                    |                                                             |  /           /     2\ |                                                                                         |    /                  /     2\                         /                  /     2\                    |                                                                                            |  /           /     2\ |                                                                                                                                                                                                                                                                                                                                                                                                           
|      |      |                                                                                                                                                                                                                                                                                                                                    2*\/  polar_lift\1 - x /                                                                        2*\1 - x /                                                                     2*x *\/  polar_lift\1 - x /                                                6*y *\/  polar_lift\1 - x /                                                      \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                             2*x *\1 - x /                                                                           \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                             \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                                \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                             \\/  polar_lift\1 - x / /                                                                                         \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                                                            \\/  polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                           
|      |      |                                                                                                                                                                                                                                                                                                           - ------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - ----------------------------------------------------------------------------------------------------------------------------                                                                                                                                                                                                                                                                                                                  otherwise                                        
|      |      |                                                                                                                                                                                                                                                                                                                       ________________________              ________________________   /          ________________________              ________________________\                                     ________________________              ________________________             ________________________              ________________________                                                                                                          2                                /          ________________________              ________________________\                                                                                                                                  2                                                                                                                                                              2                                                                                                                                                              2                                        ________________________ /          ________________________              ________________________\                                                                                                                                                    2                                                              ________________________ /          ________________________              ________________________\                                                                                                                                                                                                                                                                                                                                                                                      
|      |      |                                                                                                                                                                                                                                                                                                                      /              2                      /              2            |         /              2                      /              2         |    ____________________            /              2                      /              2                     /              2                      /              2                                         /          ________________________              ________________________\                                 |         /              2                      /              2         |    ____________________                                /          ________________________              ________________________\                                                                                     /          ________________________              ________________________\                                                                                     /          ________________________              ________________________\                                        /              2          |         /              2                      /              2         |                                                                          /          ________________________              ________________________\                                                              /              2          |         /              2                      /              2         |                                                                                                                                                                                                                                                                                                                                                                                      
|      |      |                                                                                                                                                                                                                                                                                                                     /              y                2     /              y             |        /              y                2     /              y          |   /           /     2\            /              y                2     /              y                     /              y                2     /              y                                          |         /              2                      /              2         |                                 |        /              y                2     /              y          |   /           /     2\                                 |         /              2                      /              2         |                                                                                     |         /              2                      /              2         |                                                                                     |         /              2                      /              2         |                                       /              y           |        /              y                2     /              y          |           /     2\                                                       |         /              2                      /              2         |                                                             /              y           |        /              y                2     /              y          |           /     2\                                                                                                                                                                                                                                                                                                                                                                   
|      |      |                                                                                                                                                                                                                                                                                                             - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------    |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*\/  polar_lift\1 - x /    - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------    - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------                                 |        /              y                2     /              y          |                                 |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*\/  polar_lift\1 - x /                                 |        /              y                2     /              y          |                                                                                     |        /              y                2     /              y          |                                                                                     |        /              y                2     /              y          |                                      /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*polar_lift\1 - x /                                                       |        /              y                2     /              y          |                                                            /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*polar_lift\1 - x /                                                                                                                                                                                                                                                                                                                                                                   
|      |      |                                                                                                                                                                                                                                                                                                                   /                  /     2\           /                  /     2\    |      /                  /     2\           /                  /     2\ |                                 /                  /     2\           /                  /     2\          /                  /     2\           /                  /     2\                                 |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                 |      /                  /     2\           /                  /     2\ |                                                        |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                                                                     |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                                                                     |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                     /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                          |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                                           /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                                                                                                                                                                                                                                                                                                                                      
|      |      |                                                                                                                                                                                                                                                                                                                 \/         polar_lift\1 - x /         \/         polar_lift\1 - x /    \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                               \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /                                 |      /                  /     2\           /                  /     2\ |                                 \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                        |      /                  /     2\           /                  /     2\ |                                                                                     |      /                  /     2\           /                  /     2\ |                                                                                     |      /                  /     2\           /                  /     2\ |                                   \/         polar_lift\1 - x /  \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                          |      /                  /     2\           /                  /     2\ |                                                         \/         polar_lift\1 - x /  \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                      
|      |      \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                   \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                     \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                     \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                      \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
|      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
|     /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
|    ________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
|   /      2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
|-\/  1 - x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
<                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           
|      ________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
|     /      2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
|  -\/  1 - x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
|        /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
|       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
|       |      /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           _________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             _________________________                                                                                                                                _________________________                                                                                                                                                                  _________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                               ____________________ /                                                                                2                      \             /               2                   /                                                                                2                      \            ____________________ /                                                                                2                      \               ____________________ /                                                                                2                      \                                                                                                                                                  /               2                   /                                                                                2                      \            /               2                   /                                                                                2                      \                                              /               2                   /                                                                                2                      \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                          3   /           /     2\  |                       4*y                                                 4*y*x                       |            /               y           /     2\ |                       4*y                                                 4*y*x                       |           /           /     2\  |                       4*y                                                 4*y*x                       |          2   /           /     2\  |                       4*y                                                 4*y*x                       |                                                                                                                                           2     /               y           /     2\ |                       4*y                                                 4*y*x                       |           /               y           /     2\ |                       4*y                                                 4*y*x                       |      /           y           \        2     /               y           /     2\ |                       4*y                                                 4*y*x                       |      /           y           \                                                                                                                                                                                                                                                                                                                                                                                                                                                            
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                     4*I*y *\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|   2*pi*   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|   4*I*y*\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|   4*I*y*x *\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|                                                                                                                                     2*pi*x *   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|   4*I*   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|   4*I*x *   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                            
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |      _________________________                            _________________________                   |          /                   /     2\           |      _________________________                            _________________________                   |                                 |      _________________________                            _________________________                   |                                    |      _________________________                            _________________________                   |                                                                                                                                               /                   /     2\           |      _________________________                            _________________________                   |         /                   /     2\           |      _________________________                            _________________________                   |      |   ____________________|            /                   /     2\           |      _________________________                            _________________________                   |      |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                            
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |     /               2                                    /               2                            |        \/          polar_lift\1 - x /           |     /               2                                    /               2                            |                                 |     /               2                                    /               2                            |                                    |     /               2                                    /               2                            |                                                                                                                                             \/          polar_lift\1 - x /           |     /               2                                    /               2                            |       \/          polar_lift\1 - x /           |     /               2                                    /               2                            |      |  /           /     2\ |          \/          polar_lift\1 - x /           |     /               2                                    /               2                            |      |  /           /     2\ |                                                                                                                                                                                    /     2\      /           y           \                                                                                           2 /     2\      /           y           \                                                                                             
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |    /               y                     /     2\       /               y                     /     2\|                                                 |    /               y                     /     2\       /               y                     /     2\|                                 |    /               y                     /     2\       /               y                     /     2\|                                    |    /               y                     /     2\       /               y                     /     2\|                                                                                                                                                                                      |    /               y                     /     2\       /               y                     /     2\|                                                |    /               y                     /     2\       /               y                     /     2\|      \\/  polar_lift\1 - x / /                                                   |    /               y                     /     2\       /               y                     /     2\|      \\/  polar_lift\1 - x / /                                                                                                                                                                              4*I*y*\1 - x /*acosh|-----------------------|                                                                                    4*I*y*x *\1 - x /*acosh|-----------------------|                                                                                             
|       |      |                              ____________________                                                           ____________________                                                         ____________________                                                                                                                                                                                                                                                                      |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                 |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                 |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                    |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                                                                                                                                                      |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                                                  |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                                                                                                                                                                                                 |   ____________________|                                                                                                           |   ____________________|                                                                                             
|       |      |                             /           /     2\                                                       2   /           /     2\                                                     2   /           /     2\                                                                          /     2\                                                                                              2 /     2\                                                                             |  /                   /     2\                         /                   /     2\                    |                                                 |  /                   /     2\                         /                   /     2\                    |                                 |  /                   /     2\                         /                   /     2\                    |                                    |  /                   /     2\                         /                   /     2\                    |                                                                  /     2\                                                                                                            |  /                   /     2\                         /                   /     2\                    |                                                |  /                   /     2\                         /                   /     2\                    |                                                                                  |  /                   /     2\                         /                   /     2\                    |                                                                                                 2 /     2\                                                                                                                      |  /           /     2\ |                                                                                                           |  /           /     2\ |                                             |    2  |                                       
|       |      |                       4*I*\/  polar_lift\1 - x /                                                 12*I*y *\/  polar_lift\1 - x /                                                4*I*x *\/  polar_lift\1 - x /                                                                      4*I*\1 - x /                                                                                         4*I*x *\1 - x /                                                                             \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                 \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                 \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                    \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                           2*pi*y*\1 - x /                                                                                                            \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                                                  \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                                                         2*pi*y*x *\1 - x /                                                                                                                      \\/  polar_lift\1 - x / /                                                                                                           \\/  polar_lift\1 - x / /                                             |   y   |                                       
|       |      |-------------------------------------------------------------------------- - -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------  for |-------| > 1                                   
|       |      |          _________________________              _________________________             _________________________              _________________________             _________________________              _________________________   /          _________________________              _________________________\                           /          _________________________              _________________________\                                                                                                                                    2                                                                                                                                                  2                                                                                                                                                 2                                                                                                                                          2                                        _________________________ /          _________________________              _________________________\                                                                                                                                        2                                                                                                                                                                          2                                                                                                                                                                                        2                                                               _________________________ /          _________________________              _________________________\                            _________________________ /          _________________________              _________________________\                            _________________________ /          _________________________              _________________________\                         |      2|                                       
|       |      |         /               2                      /               2                     /               2                      /               2                     /               2                      /               2            |         /               2                      /               2         |    ____________________   |         /               2                      /               2         |    ____________________                                /          _________________________              _________________________\                                                                       /          _________________________              _________________________\                                                                      /          _________________________              _________________________\                                                               /          _________________________              _________________________\                                        /               2          |         /               2                      /               2         |                                                            /          _________________________              _________________________\                                                                                               /          _________________________              _________________________\                                                                                                             /          _________________________              _________________________\                                                               /               2          |         /               2                      /               2         |                           /               2          |         /               2                      /               2         |                           /               2          |         /               2                      /               2         |                         |-1 + x |                                       
|       |      |        /               y                2     /               y                     /               y                2     /               y                     /               y                2     /               y             |        /               y                2     /               y          |   /           /     2\    |        /               y                2     /               y          |   /           /     2\                                 |         /               2                      /               2         |                                                                       |         /               2                      /               2         |                                                                      |         /               2                      /               2         |                                                               |         /               2                      /               2         |                                       /               y           |        /               y                2     /               y          |           /     2\                                         |         /               2                      /               2         |                                                                                               |         /               2                      /               2         |                                                                                                             |         /               2                      /               2         |                                                              /               y           |        /               y                2     /               y          |           /     2\       /               y           |        /               y                2     /               y          |           /     2\       /               y           |        /               y                2     /               y          |           /     2\                                                      
|       |      |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*\/  polar_lift\1 - x /    |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*\/  polar_lift\1 - x /                                 |        /               y                2     /               y          |                                                                       |        /               y                2     /               y          |                                                                      |        /               y                2     /               y          |                                                               |        /               y                2     /               y          |                                      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /                                         |        /               y                2     /               y          |                                                                                               |        /               y                2     /               y          |                                                                                                             |        /               y                2     /               y          |                                                             /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /                                                      
|       |      |      /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\    |      /                   /     2\           /                   /     2\ |                           |      /                   /     2\           /                   /     2\ |                                                        |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                       |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                      |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                               |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                     /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                            |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                                               |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                                                             |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                            /                   /     2\  |      /                   /     2\           /                   /     2\ |                        /                   /     2\  |      /                   /     2\           /                   /     2\ |                        /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                                         
|       |      |    \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /    \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                           \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                        |      /                   /     2\           /                   /     2\ |                                                                       |      /                   /     2\           /                   /     2\ |                                                                      |      /                   /     2\           /                   /     2\ |                                                               |      /                   /     2\           /                   /     2\ |                                   \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                            |      /                   /     2\           /                   /     2\ |                                                                                               |      /                   /     2\           /                   /     2\ |                                                                                                             |      /                   /     2\           /                   /     2\ |                                                          \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                      \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                      \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                         
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                       \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                      \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                               \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                                                                                                                           \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                               \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                             \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
|-      |      <                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          dy            otherwise           
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 ________________________                                                                                                                                                                                                                                                                                                                                                                                                                                         ________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          ____________________ /                                                                                 2                     \                                                                                                                ____________________ /                                                                                 2                     \          /              2                   /                                                                                 2                     \                                           ____________________ /                                                                                 2                     \                                                                                                                                            /              2                   /                                                                                 2                     \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         /           /     2\  |                         2*y                                                2*y*x                      |                                                                                                           3   /           /     2\  |                         2*y                                                2*y*x                      |         /              y           /     2\ |                         2*y                                                2*y*x                      |     /           y           \        2   /           /     2\  |                         2*y                                                2*y*x                      |                                                                                                                                     2     /              y           /     2\ |                         2*y                                                2*y*x                      |     /           y           \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   2*y*\/  polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------|                                                                                                        2*y *\/  polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------|   2*   /   1 - ------------------ *\1 - x /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|   2*y*x *\/  polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------|                                                                                                                                  2*x *   /   1 - ------------------ *\1 - x /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |        ________________________                            ________________________                   |                                                                                                                                     |        ________________________                            ________________________                   |       /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|                                  |        ________________________                            ________________________                   |                                                                                                                                         /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |       /              2                                    /              2                            |                                                                                                                                     |       /              2                                    /              2                            |     \/         polar_lift\1 - x /           |       /              2                                    /              2                            |     |  /           /     2\ |                                  |       /              2                                    /              2                            |                                                /     2\     /           y           \                                                 \/         polar_lift\1 - x /           |       /              2                                    /              2                            |     |  /           /     2\ |                                               2 /     2\     /           y           \                                                                                                                                                                                                                                                                                                                                                                                                          
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |      /              y                     /     2\       /              y                     /     2\|                                                                                                                                     |      /              y                     /     2\       /              y                     /     2\|                                             |      /              y                     /     2\       /              y                     /     2\|     \\/  polar_lift\1 - x / /                                  |      /              y                     /     2\       /              y                     /     2\|                                            2*y*\1 - x /*asin|-----------------------|                                                                                         |      /              y                     /     2\       /              y                     /     2\|     \\/  polar_lift\1 - x / /                                          2*y*x *\1 - x /*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                          
|       |      |                                                                                                                                                                                                                                                                                                                                         ____________________                                                                                                                                                              ____________________                                                       ____________________                                                     |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                                                                                                     |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                             |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                                |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                             |   ____________________|                                                                                         |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                                                            |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                          
|       |      |                                                                                                                                                                                                                                                                                                                                        /           /     2\                                                                          /     2\                                                                        2   /           /     2\                                                   2   /           /     2\                                                      |    /                  /     2\                         /                  /     2\                    |                                                2 /     2\                                                                           |    /                  /     2\                         /                  /     2\                    |                                             |    /                  /     2\                         /                  /     2\                    |                                                                |    /                  /     2\                         /                  /     2\                    |                                                             |  /           /     2\ |                                                                                         |    /                  /     2\                         /                  /     2\                    |                                                                                            |  /           /     2\ |                                                                                                                                                                                                                                                                                                                                                                                                          
|       |      |                                                                                                                                                                                                                                                                                                                                    2*\/  polar_lift\1 - x /                                                                        2*\1 - x /                                                                     2*x *\/  polar_lift\1 - x /                                                6*y *\/  polar_lift\1 - x /                                                      \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                             2*x *\1 - x /                                                                           \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                             \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                                \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                             \\/  polar_lift\1 - x / /                                                                                         \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                                                            \\/  polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                          
|       |      |                                                                                                                                                                                                                                                                                                           - ------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - ----------------------------------------------------------------------------------------------------------------------------                                                                                                                                                                                                                                                                                                                  otherwise                                       
|       |      |                                                                                                                                                                                                                                                                                                                       ________________________              ________________________   /          ________________________              ________________________\                                     ________________________              ________________________             ________________________              ________________________                                                                                                          2                                /          ________________________              ________________________\                                                                                                                                  2                                                                                                                                                              2                                                                                                                                                              2                                        ________________________ /          ________________________              ________________________\                                                                                                                                                    2                                                              ________________________ /          ________________________              ________________________\                                                                                                                                                                                                                                                                                                                                                                                     
|       |      |                                                                                                                                                                                                                                                                                                                      /              2                      /              2            |         /              2                      /              2         |    ____________________            /              2                      /              2                     /              2                      /              2                                         /          ________________________              ________________________\                                 |         /              2                      /              2         |    ____________________                                /          ________________________              ________________________\                                                                                     /          ________________________              ________________________\                                                                                     /          ________________________              ________________________\                                        /              2          |         /              2                      /              2         |                                                                          /          ________________________              ________________________\                                                              /              2          |         /              2                      /              2         |                                                                                                                                                                                                                                                                                                                                                                                     
|       |      |                                                                                                                                                                                                                                                                                                                     /              y                2     /              y             |        /              y                2     /              y          |   /           /     2\            /              y                2     /              y                     /              y                2     /              y                                          |         /              2                      /              2         |                                 |        /              y                2     /              y          |   /           /     2\                                 |         /              2                      /              2         |                                                                                     |         /              2                      /              2         |                                                                                     |         /              2                      /              2         |                                       /              y           |        /              y                2     /              y          |           /     2\                                                       |         /              2                      /              2         |                                                             /              y           |        /              y                2     /              y          |           /     2\                                                                                                                                                                                                                                                                                                                                                                  
|       |      |                                                                                                                                                                                                                                                                                                             - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------    |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*\/  polar_lift\1 - x /    - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------    - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------                                 |        /              y                2     /              y          |                                 |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*\/  polar_lift\1 - x /                                 |        /              y                2     /              y          |                                                                                     |        /              y                2     /              y          |                                                                                     |        /              y                2     /              y          |                                      /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*polar_lift\1 - x /                                                       |        /              y                2     /              y          |                                                            /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*polar_lift\1 - x /                                                                                                                                                                                                                                                                                                                                                                  
|       |      |                                                                                                                                                                                                                                                                                                                   /                  /     2\           /                  /     2\    |      /                  /     2\           /                  /     2\ |                                 /                  /     2\           /                  /     2\          /                  /     2\           /                  /     2\                                 |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                 |      /                  /     2\           /                  /     2\ |                                                        |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                                                                     |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                                                                     |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                     /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                          |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                                           /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                                                                                                                                                                                                                                                                                                                                     
|       |      |                                                                                                                                                                                                                                                                                                                 \/         polar_lift\1 - x /         \/         polar_lift\1 - x /    \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                               \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /                                 |      /                  /     2\           /                  /     2\ |                                 \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                        |      /                  /     2\           /                  /     2\ |                                                                                     |      /                  /     2\           /                  /     2\ |                                                                                     |      /                  /     2\           /                  /     2\ |                                   \/         polar_lift\1 - x /  \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                          |      /                  /     2\           /                  /     2\ |                                                         \/         polar_lift\1 - x /  \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                     
|       |      \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                   \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                     \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                     \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                      \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
|       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
|      /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
|     ________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
|    /      2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
\  \/  1 - x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
$$\begin{cases} \int\limits_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} \begin{cases} - \frac{4 i x^{2} y \left(1 - x^{2}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{2 \pi x^{2} y \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{4 i x^{2} y \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{4 i x^{2} \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} - \frac{4 i x^{2} \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} + \frac{2 \pi x^{2} \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{4 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} - \frac{4 i y^{3} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{12 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} + \frac{4 i y \left(1 - x^{2}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 \pi y \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 i y \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} + \frac{4 i \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} + \frac{4 i \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{2 \pi \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} + \frac{4 i \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} & \text{for}\: \left|{\frac{y^{2}}{x^{2} - 1}}\right| > 1 \\- \frac{2 x^{2} y \left(1 - x^{2}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{2 x^{2} y \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} + \frac{2 x^{2} \left(1 - x^{2}\right)}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} + \frac{2 x^{2} \left(1 - x^{2}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} + \frac{2 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} + \frac{2 y^{3} \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} + \frac{6 y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} + \frac{2 y \left(1 - x^{2}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} - \frac{2 \left(1 - x^{2}\right)}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} - \frac{2 \left(1 - x^{2}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} - \frac{2 \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dy & \text{for}\: \sqrt{1 - x^{2}} > - \sqrt{1 - x^{2}} \\- \int\limits_{\sqrt{1 - x^{2}}}^{- \sqrt{1 - x^{2}}} \begin{cases} - \frac{4 i x^{2} y \left(1 - x^{2}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{2 \pi x^{2} y \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{4 i x^{2} y \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{4 i x^{2} \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} - \frac{4 i x^{2} \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} + \frac{2 \pi x^{2} \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{4 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} - \frac{4 i y^{3} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{12 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} + \frac{4 i y \left(1 - x^{2}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 \pi y \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 i y \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} + \frac{4 i \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} + \frac{4 i \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{2 \pi \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} + \frac{4 i \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} & \text{for}\: \left|{\frac{y^{2}}{x^{2} - 1}}\right| > 1 \\- \frac{2 x^{2} y \left(1 - x^{2}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{2 x^{2} y \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} + \frac{2 x^{2} \left(1 - x^{2}\right)}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} + \frac{2 x^{2} \left(1 - x^{2}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} + \frac{2 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} + \frac{2 y^{3} \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} + \frac{6 y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} + \frac{2 y \left(1 - x^{2}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} - \frac{2 \left(1 - x^{2}\right)}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} - \frac{2 \left(1 - x^{2}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} - \frac{2 \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dy & \text{otherwise} \end{cases}$$
=
=
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|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                          3   /           /     2\  |                       4*y                                                 4*y*x                       |            /               y           /     2\ |                       4*y                                                 4*y*x                       |           /           /     2\  |                       4*y                                                 4*y*x                       |          2   /           /     2\  |                       4*y                                                 4*y*x                       |                                                                                                                                           2     /               y           /     2\ |                       4*y                                                 4*y*x                       |           /               y           /     2\ |                       4*y                                                 4*y*x                       |      /           y           \        2     /               y           /     2\ |                       4*y                                                 4*y*x                       |      /           y           \                                                                                                                                                                                                                                                                                                                                                                                                                                                             
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                     4*I*y *\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|   2*pi*   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|   4*I*y*\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|   4*I*y*x *\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|                                                                                                                                     2*pi*x *   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|   4*I*   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|   4*I*x *   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                             
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |      _________________________                            _________________________                   |          /                   /     2\           |      _________________________                            _________________________                   |                                 |      _________________________                            _________________________                   |                                    |      _________________________                            _________________________                   |                                                                                                                                               /                   /     2\           |      _________________________                            _________________________                   |         /                   /     2\           |      _________________________                            _________________________                   |      |   ____________________|            /                   /     2\           |      _________________________                            _________________________                   |      |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                             
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |     /               2                                    /               2                            |        \/          polar_lift\1 - x /           |     /               2                                    /               2                            |                                 |     /               2                                    /               2                            |                                    |     /               2                                    /               2                            |                                                                                                                                             \/          polar_lift\1 - x /           |     /               2                                    /               2                            |       \/          polar_lift\1 - x /           |     /               2                                    /               2                            |      |  /           /     2\ |          \/          polar_lift\1 - x /           |     /               2                                    /               2                            |      |  /           /     2\ |                                                                                                                                                                                    /     2\      /           y           \                                                                                           2 /     2\      /           y           \                                                                                              
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|      |      |                              ____________________                                                           ____________________                                                         ____________________                                                                                                                                                                                                                                                                      |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                 |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                 |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                    |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                                                                                                                                                      |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                                                  |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                                                                                                                                                                                                 |   ____________________|                                                                                                           |   ____________________|                                                                                              
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|      |      |                       4*I*\/  polar_lift\1 - x /                                                 12*I*y *\/  polar_lift\1 - x /                                                4*I*x *\/  polar_lift\1 - x /                                                                      4*I*\1 - x /                                                                                         4*I*x *\1 - x /                                                                             \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                 \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                 \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                    \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                           2*pi*y*\1 - x /                                                                                                            \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                                                  \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                                                         2*pi*y*x *\1 - x /                                                                                                                      \\/  polar_lift\1 - x / /                                                                                                           \\/  polar_lift\1 - x / /                                             |   y   |                                        
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|      |      |          _________________________              _________________________             _________________________              _________________________             _________________________              _________________________   /          _________________________              _________________________\                           /          _________________________              _________________________\                                                                                                                                    2                                                                                                                                                  2                                                                                                                                                 2                                                                                                                                          2                                        _________________________ /          _________________________              _________________________\                                                                                                                                        2                                                                                                                                                                          2                                                                                                                                                                                        2                                                               _________________________ /          _________________________              _________________________\                            _________________________ /          _________________________              _________________________\                            _________________________ /          _________________________              _________________________\                         |      2|                                        
|      |      |         /               2                      /               2                     /               2                      /               2                     /               2                      /               2            |         /               2                      /               2         |    ____________________   |         /               2                      /               2         |    ____________________                                /          _________________________              _________________________\                                                                       /          _________________________              _________________________\                                                                      /          _________________________              _________________________\                                                               /          _________________________              _________________________\                                        /               2          |         /               2                      /               2         |                                                            /          _________________________              _________________________\                                                                                               /          _________________________              _________________________\                                                                                                             /          _________________________              _________________________\                                                               /               2          |         /               2                      /               2         |                           /               2          |         /               2                      /               2         |                           /               2          |         /               2                      /               2         |                         |-1 + x |                                        
|      |      |        /               y                2     /               y                     /               y                2     /               y                     /               y                2     /               y             |        /               y                2     /               y          |   /           /     2\    |        /               y                2     /               y          |   /           /     2\                                 |         /               2                      /               2         |                                                                       |         /               2                      /               2         |                                                                      |         /               2                      /               2         |                                                               |         /               2                      /               2         |                                       /               y           |        /               y                2     /               y          |           /     2\                                         |         /               2                      /               2         |                                                                                               |         /               2                      /               2         |                                                                                                             |         /               2                      /               2         |                                                              /               y           |        /               y                2     /               y          |           /     2\       /               y           |        /               y                2     /               y          |           /     2\       /               y           |        /               y                2     /               y          |           /     2\                                                       
|      |      |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*\/  polar_lift\1 - x /    |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*\/  polar_lift\1 - x /                                 |        /               y                2     /               y          |                                                                       |        /               y                2     /               y          |                                                                      |        /               y                2     /               y          |                                                               |        /               y                2     /               y          |                                      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /                                         |        /               y                2     /               y          |                                                                                               |        /               y                2     /               y          |                                                                                                             |        /               y                2     /               y          |                                                             /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /                                                       
|      |      |      /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\    |      /                   /     2\           /                   /     2\ |                           |      /                   /     2\           /                   /     2\ |                                                        |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                       |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                      |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                               |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                     /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                            |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                                               |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                                                             |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                            /                   /     2\  |      /                   /     2\           /                   /     2\ |                        /                   /     2\  |      /                   /     2\           /                   /     2\ |                        /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                                          
|      |      |    \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /    \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                           \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                        |      /                   /     2\           /                   /     2\ |                                                                       |      /                   /     2\           /                   /     2\ |                                                                      |      /                   /     2\           /                   /     2\ |                                                               |      /                   /     2\           /                   /     2\ |                                   \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                            |      /                   /     2\           /                   /     2\ |                                                                                               |      /                   /     2\           /                   /     2\ |                                                                                                             |      /                   /     2\           /                   /     2\ |                                                          \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                      \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                      \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                   ________       ________
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                       \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                      \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                               \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                                                                                                                           \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                               \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                             \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            /      2       /      2 
|      |      <                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          dy   for \/  1 - x   > -\/  1 - x  
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 ________________________                                                                                                                                                                                                                                                                                                                                                                                                                                         ________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          ____________________ /                                                                                 2                     \                                                                                                                ____________________ /                                                                                 2                     \          /              2                   /                                                                                 2                     \                                           ____________________ /                                                                                 2                     \                                                                                                                                            /              2                   /                                                                                 2                     \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         /           /     2\  |                         2*y                                                2*y*x                      |                                                                                                           3   /           /     2\  |                         2*y                                                2*y*x                      |         /              y           /     2\ |                         2*y                                                2*y*x                      |     /           y           \        2   /           /     2\  |                         2*y                                                2*y*x                      |                                                                                                                                     2     /              y           /     2\ |                         2*y                                                2*y*x                      |     /           y           \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   2*y*\/  polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------|                                                                                                        2*y *\/  polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------|   2*   /   1 - ------------------ *\1 - x /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|   2*y*x *\/  polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------|                                                                                                                                  2*x *   /   1 - ------------------ *\1 - x /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |        ________________________                            ________________________                   |                                                                                                                                     |        ________________________                            ________________________                   |       /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|                                  |        ________________________                            ________________________                   |                                                                                                                                         /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |       /              2                                    /              2                            |                                                                                                                                     |       /              2                                    /              2                            |     \/         polar_lift\1 - x /           |       /              2                                    /              2                            |     |  /           /     2\ |                                  |       /              2                                    /              2                            |                                                /     2\     /           y           \                                                 \/         polar_lift\1 - x /           |       /              2                                    /              2                            |     |  /           /     2\ |                                               2 /     2\     /           y           \                                                                                                                                                                                                                                                                                                                                                                                                           
|      |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |      /              y                     /     2\       /              y                     /     2\|                                                                                                                                     |      /              y                     /     2\       /              y                     /     2\|                                             |      /              y                     /     2\       /              y                     /     2\|     \\/  polar_lift\1 - x / /                                  |      /              y                     /     2\       /              y                     /     2\|                                            2*y*\1 - x /*asin|-----------------------|                                                                                         |      /              y                     /     2\       /              y                     /     2\|     \\/  polar_lift\1 - x / /                                          2*y*x *\1 - x /*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                           
|      |      |                                                                                                                                                                                                                                                                                                                                         ____________________                                                                                                                                                              ____________________                                                       ____________________                                                     |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                                                                                                     |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                             |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                                |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                             |   ____________________|                                                                                         |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                                                            |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                           
|      |      |                                                                                                                                                                                                                                                                                                                                        /           /     2\                                                                          /     2\                                                                        2   /           /     2\                                                   2   /           /     2\                                                      |    /                  /     2\                         /                  /     2\                    |                                                2 /     2\                                                                           |    /                  /     2\                         /                  /     2\                    |                                             |    /                  /     2\                         /                  /     2\                    |                                                                |    /                  /     2\                         /                  /     2\                    |                                                             |  /           /     2\ |                                                                                         |    /                  /     2\                         /                  /     2\                    |                                                                                            |  /           /     2\ |                                                                                                                                                                                                                                                                                                                                                                                                           
|      |      |                                                                                                                                                                                                                                                                                                                                    2*\/  polar_lift\1 - x /                                                                        2*\1 - x /                                                                     2*x *\/  polar_lift\1 - x /                                                6*y *\/  polar_lift\1 - x /                                                      \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                             2*x *\1 - x /                                                                           \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                             \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                                \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                             \\/  polar_lift\1 - x / /                                                                                         \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                                                            \\/  polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                           
|      |      |                                                                                                                                                                                                                                                                                                           - ------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - ----------------------------------------------------------------------------------------------------------------------------                                                                                                                                                                                                                                                                                                                  otherwise                                        
|      |      |                                                                                                                                                                                                                                                                                                                       ________________________              ________________________   /          ________________________              ________________________\                                     ________________________              ________________________             ________________________              ________________________                                                                                                          2                                /          ________________________              ________________________\                                                                                                                                  2                                                                                                                                                              2                                                                                                                                                              2                                        ________________________ /          ________________________              ________________________\                                                                                                                                                    2                                                              ________________________ /          ________________________              ________________________\                                                                                                                                                                                                                                                                                                                                                                                      
|      |      |                                                                                                                                                                                                                                                                                                                      /              2                      /              2            |         /              2                      /              2         |    ____________________            /              2                      /              2                     /              2                      /              2                                         /          ________________________              ________________________\                                 |         /              2                      /              2         |    ____________________                                /          ________________________              ________________________\                                                                                     /          ________________________              ________________________\                                                                                     /          ________________________              ________________________\                                        /              2          |         /              2                      /              2         |                                                                          /          ________________________              ________________________\                                                              /              2          |         /              2                      /              2         |                                                                                                                                                                                                                                                                                                                                                                                      
|      |      |                                                                                                                                                                                                                                                                                                                     /              y                2     /              y             |        /              y                2     /              y          |   /           /     2\            /              y                2     /              y                     /              y                2     /              y                                          |         /              2                      /              2         |                                 |        /              y                2     /              y          |   /           /     2\                                 |         /              2                      /              2         |                                                                                     |         /              2                      /              2         |                                                                                     |         /              2                      /              2         |                                       /              y           |        /              y                2     /              y          |           /     2\                                                       |         /              2                      /              2         |                                                             /              y           |        /              y                2     /              y          |           /     2\                                                                                                                                                                                                                                                                                                                                                                   
|      |      |                                                                                                                                                                                                                                                                                                             - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------    |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*\/  polar_lift\1 - x /    - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------    - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------                                 |        /              y                2     /              y          |                                 |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*\/  polar_lift\1 - x /                                 |        /              y                2     /              y          |                                                                                     |        /              y                2     /              y          |                                                                                     |        /              y                2     /              y          |                                      /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*polar_lift\1 - x /                                                       |        /              y                2     /              y          |                                                            /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*polar_lift\1 - x /                                                                                                                                                                                                                                                                                                                                                                   
|      |      |                                                                                                                                                                                                                                                                                                                   /                  /     2\           /                  /     2\    |      /                  /     2\           /                  /     2\ |                                 /                  /     2\           /                  /     2\          /                  /     2\           /                  /     2\                                 |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                 |      /                  /     2\           /                  /     2\ |                                                        |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                                                                     |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                                                                     |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                     /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                          |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                                           /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                                                                                                                                                                                                                                                                                                                                      
|      |      |                                                                                                                                                                                                                                                                                                                 \/         polar_lift\1 - x /         \/         polar_lift\1 - x /    \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                               \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /                                 |      /                  /     2\           /                  /     2\ |                                 \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                        |      /                  /     2\           /                  /     2\ |                                                                                     |      /                  /     2\           /                  /     2\ |                                                                                     |      /                  /     2\           /                  /     2\ |                                   \/         polar_lift\1 - x /  \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                          |      /                  /     2\           /                  /     2\ |                                                         \/         polar_lift\1 - x /  \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                      
|      |      \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                   \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                     \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                     \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                      \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
|      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
|     /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
|    ________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
|   /      2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
|-\/  1 - x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
<                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           
|      ________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
|     /      2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
|  -\/  1 - x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
|        /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
|       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
|       |      /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           _________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             _________________________                                                                                                                                _________________________                                                                                                                                                                  _________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                               ____________________ /                                                                                2                      \             /               2                   /                                                                                2                      \            ____________________ /                                                                                2                      \               ____________________ /                                                                                2                      \                                                                                                                                                  /               2                   /                                                                                2                      \            /               2                   /                                                                                2                      \                                              /               2                   /                                                                                2                      \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                          3   /           /     2\  |                       4*y                                                 4*y*x                       |            /               y           /     2\ |                       4*y                                                 4*y*x                       |           /           /     2\  |                       4*y                                                 4*y*x                       |          2   /           /     2\  |                       4*y                                                 4*y*x                       |                                                                                                                                           2     /               y           /     2\ |                       4*y                                                 4*y*x                       |           /               y           /     2\ |                       4*y                                                 4*y*x                       |      /           y           \        2     /               y           /     2\ |                       4*y                                                 4*y*x                       |      /           y           \                                                                                                                                                                                                                                                                                                                                                                                                                                                            
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                     4*I*y *\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|   2*pi*   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|   4*I*y*\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|   4*I*y*x *\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|                                                                                                                                     2*pi*x *   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|   4*I*   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|   4*I*x *   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                            
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |      _________________________                            _________________________                   |          /                   /     2\           |      _________________________                            _________________________                   |                                 |      _________________________                            _________________________                   |                                    |      _________________________                            _________________________                   |                                                                                                                                               /                   /     2\           |      _________________________                            _________________________                   |         /                   /     2\           |      _________________________                            _________________________                   |      |   ____________________|            /                   /     2\           |      _________________________                            _________________________                   |      |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                            
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |     /               2                                    /               2                            |        \/          polar_lift\1 - x /           |     /               2                                    /               2                            |                                 |     /               2                                    /               2                            |                                    |     /               2                                    /               2                            |                                                                                                                                             \/          polar_lift\1 - x /           |     /               2                                    /               2                            |       \/          polar_lift\1 - x /           |     /               2                                    /               2                            |      |  /           /     2\ |          \/          polar_lift\1 - x /           |     /               2                                    /               2                            |      |  /           /     2\ |                                                                                                                                                                                    /     2\      /           y           \                                                                                           2 /     2\      /           y           \                                                                                             
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |    /               y                     /     2\       /               y                     /     2\|                                                 |    /               y                     /     2\       /               y                     /     2\|                                 |    /               y                     /     2\       /               y                     /     2\|                                    |    /               y                     /     2\       /               y                     /     2\|                                                                                                                                                                                      |    /               y                     /     2\       /               y                     /     2\|                                                |    /               y                     /     2\       /               y                     /     2\|      \\/  polar_lift\1 - x / /                                                   |    /               y                     /     2\       /               y                     /     2\|      \\/  polar_lift\1 - x / /                                                                                                                                                                              4*I*y*\1 - x /*acosh|-----------------------|                                                                                    4*I*y*x *\1 - x /*acosh|-----------------------|                                                                                             
|       |      |                              ____________________                                                           ____________________                                                         ____________________                                                                                                                                                                                                                                                                      |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                 |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                 |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                    |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                                                                                                                                                      |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                                                  |   /   -1 + ------------------ *polar_lift\1 - x /      /   -1 + ------------------ *polar_lift\1 - x /|                                                                                                                                                                                                                                 |   ____________________|                                                                                                           |   ____________________|                                                                                             
|       |      |                             /           /     2\                                                       2   /           /     2\                                                     2   /           /     2\                                                                          /     2\                                                                                              2 /     2\                                                                             |  /                   /     2\                         /                   /     2\                    |                                                 |  /                   /     2\                         /                   /     2\                    |                                 |  /                   /     2\                         /                   /     2\                    |                                    |  /                   /     2\                         /                   /     2\                    |                                                                  /     2\                                                                                                            |  /                   /     2\                         /                   /     2\                    |                                                |  /                   /     2\                         /                   /     2\                    |                                                                                  |  /                   /     2\                         /                   /     2\                    |                                                                                                 2 /     2\                                                                                                                      |  /           /     2\ |                                                                                                           |  /           /     2\ |                                             |    2  |                                       
|       |      |                       4*I*\/  polar_lift\1 - x /                                                 12*I*y *\/  polar_lift\1 - x /                                                4*I*x *\/  polar_lift\1 - x /                                                                      4*I*\1 - x /                                                                                         4*I*x *\1 - x /                                                                             \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                 \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                 \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                    \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                           2*pi*y*\1 - x /                                                                                                            \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                                                  \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                                                         2*pi*y*x *\1 - x /                                                                                                                      \\/  polar_lift\1 - x / /                                                                                                           \\/  polar_lift\1 - x / /                                             |   y   |                                       
|       |      |-------------------------------------------------------------------------- - -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------  for |-------| > 1                                   
|       |      |          _________________________              _________________________             _________________________              _________________________             _________________________              _________________________   /          _________________________              _________________________\                           /          _________________________              _________________________\                                                                                                                                    2                                                                                                                                                  2                                                                                                                                                 2                                                                                                                                          2                                        _________________________ /          _________________________              _________________________\                                                                                                                                        2                                                                                                                                                                          2                                                                                                                                                                                        2                                                               _________________________ /          _________________________              _________________________\                            _________________________ /          _________________________              _________________________\                            _________________________ /          _________________________              _________________________\                         |      2|                                       
|       |      |         /               2                      /               2                     /               2                      /               2                     /               2                      /               2            |         /               2                      /               2         |    ____________________   |         /               2                      /               2         |    ____________________                                /          _________________________              _________________________\                                                                       /          _________________________              _________________________\                                                                      /          _________________________              _________________________\                                                               /          _________________________              _________________________\                                        /               2          |         /               2                      /               2         |                                                            /          _________________________              _________________________\                                                                                               /          _________________________              _________________________\                                                                                                             /          _________________________              _________________________\                                                               /               2          |         /               2                      /               2         |                           /               2          |         /               2                      /               2         |                           /               2          |         /               2                      /               2         |                         |-1 + x |                                       
|       |      |        /               y                2     /               y                     /               y                2     /               y                     /               y                2     /               y             |        /               y                2     /               y          |   /           /     2\    |        /               y                2     /               y          |   /           /     2\                                 |         /               2                      /               2         |                                                                       |         /               2                      /               2         |                                                                      |         /               2                      /               2         |                                                               |         /               2                      /               2         |                                       /               y           |        /               y                2     /               y          |           /     2\                                         |         /               2                      /               2         |                                                                                               |         /               2                      /               2         |                                                                                                             |         /               2                      /               2         |                                                              /               y           |        /               y                2     /               y          |           /     2\       /               y           |        /               y                2     /               y          |           /     2\       /               y           |        /               y                2     /               y          |           /     2\                                                      
|       |      |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*\/  polar_lift\1 - x /    |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*\/  polar_lift\1 - x /                                 |        /               y                2     /               y          |                                                                       |        /               y                2     /               y          |                                                                      |        /               y                2     /               y          |                                                               |        /               y                2     /               y          |                                      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /                                         |        /               y                2     /               y          |                                                                                               |        /               y                2     /               y          |                                                                                                             |        /               y                2     /               y          |                                                             /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /                                                      
|       |      |      /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\    |      /                   /     2\           /                   /     2\ |                           |      /                   /     2\           /                   /     2\ |                                                        |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                       |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                      |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                               |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                     /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                            |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                                               |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                                                             |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                            /                   /     2\  |      /                   /     2\           /                   /     2\ |                        /                   /     2\  |      /                   /     2\           /                   /     2\ |                        /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                                         
|       |      |    \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /    \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                           \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                        |      /                   /     2\           /                   /     2\ |                                                                       |      /                   /     2\           /                   /     2\ |                                                                      |      /                   /     2\           /                   /     2\ |                                                               |      /                   /     2\           /                   /     2\ |                                   \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                            |      /                   /     2\           /                   /     2\ |                                                                                               |      /                   /     2\           /                   /     2\ |                                                                                                             |      /                   /     2\           /                   /     2\ |                                                          \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                      \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                      \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                         
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                       \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                      \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                               \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                                                                                                                           \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                               \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                             \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
|-      |      <                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          dy            otherwise           
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 ________________________                                                                                                                                                                                                                                                                                                                                                                                                                                         ________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          ____________________ /                                                                                 2                     \                                                                                                                ____________________ /                                                                                 2                     \          /              2                   /                                                                                 2                     \                                           ____________________ /                                                                                 2                     \                                                                                                                                            /              2                   /                                                                                 2                     \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         /           /     2\  |                         2*y                                                2*y*x                      |                                                                                                           3   /           /     2\  |                         2*y                                                2*y*x                      |         /              y           /     2\ |                         2*y                                                2*y*x                      |     /           y           \        2   /           /     2\  |                         2*y                                                2*y*x                      |                                                                                                                                     2     /              y           /     2\ |                         2*y                                                2*y*x                      |     /           y           \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   2*y*\/  polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------|                                                                                                        2*y *\/  polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------|   2*   /   1 - ------------------ *\1 - x /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|   2*y*x *\/  polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------|                                                                                                                                  2*x *   /   1 - ------------------ *\1 - x /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |        ________________________                            ________________________                   |                                                                                                                                     |        ________________________                            ________________________                   |       /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|                                  |        ________________________                            ________________________                   |                                                                                                                                         /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |       /              2                                    /              2                            |                                                                                                                                     |       /              2                                    /              2                            |     \/         polar_lift\1 - x /           |       /              2                                    /              2                            |     |  /           /     2\ |                                  |       /              2                                    /              2                            |                                                /     2\     /           y           \                                                 \/         polar_lift\1 - x /           |       /              2                                    /              2                            |     |  /           /     2\ |                                               2 /     2\     /           y           \                                                                                                                                                                                                                                                                                                                                                                                                          
|       |      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |      /              y                     /     2\       /              y                     /     2\|                                                                                                                                     |      /              y                     /     2\       /              y                     /     2\|                                             |      /              y                     /     2\       /              y                     /     2\|     \\/  polar_lift\1 - x / /                                  |      /              y                     /     2\       /              y                     /     2\|                                            2*y*\1 - x /*asin|-----------------------|                                                                                         |      /              y                     /     2\       /              y                     /     2\|     \\/  polar_lift\1 - x / /                                          2*y*x *\1 - x /*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                          
|       |      |                                                                                                                                                                                                                                                                                                                                         ____________________                                                                                                                                                              ____________________                                                       ____________________                                                     |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                                                                                                     |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                             |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                                |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                             |   ____________________|                                                                                         |     /   1 - ------------------ *polar_lift\1 - x /      /   1 - ------------------ *polar_lift\1 - x /|                                                                                            |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                          
|       |      |                                                                                                                                                                                                                                                                                                                                        /           /     2\                                                                          /     2\                                                                        2   /           /     2\                                                   2   /           /     2\                                                      |    /                  /     2\                         /                  /     2\                    |                                                2 /     2\                                                                           |    /                  /     2\                         /                  /     2\                    |                                             |    /                  /     2\                         /                  /     2\                    |                                                                |    /                  /     2\                         /                  /     2\                    |                                                             |  /           /     2\ |                                                                                         |    /                  /     2\                         /                  /     2\                    |                                                                                            |  /           /     2\ |                                                                                                                                                                                                                                                                                                                                                                                                          
|       |      |                                                                                                                                                                                                                                                                                                                                    2*\/  polar_lift\1 - x /                                                                        2*\1 - x /                                                                     2*x *\/  polar_lift\1 - x /                                                6*y *\/  polar_lift\1 - x /                                                      \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                             2*x *\1 - x /                                                                           \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                             \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                                \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                             \\/  polar_lift\1 - x / /                                                                                         \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                                                            \\/  polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                          
|       |      |                                                                                                                                                                                                                                                                                                           - ------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - ----------------------------------------------------------------------------------------------------------------------------                                                                                                                                                                                                                                                                                                                  otherwise                                       
|       |      |                                                                                                                                                                                                                                                                                                                       ________________________              ________________________   /          ________________________              ________________________\                                     ________________________              ________________________             ________________________              ________________________                                                                                                          2                                /          ________________________              ________________________\                                                                                                                                  2                                                                                                                                                              2                                                                                                                                                              2                                        ________________________ /          ________________________              ________________________\                                                                                                                                                    2                                                              ________________________ /          ________________________              ________________________\                                                                                                                                                                                                                                                                                                                                                                                     
|       |      |                                                                                                                                                                                                                                                                                                                      /              2                      /              2            |         /              2                      /              2         |    ____________________            /              2                      /              2                     /              2                      /              2                                         /          ________________________              ________________________\                                 |         /              2                      /              2         |    ____________________                                /          ________________________              ________________________\                                                                                     /          ________________________              ________________________\                                                                                     /          ________________________              ________________________\                                        /              2          |         /              2                      /              2         |                                                                          /          ________________________              ________________________\                                                              /              2          |         /              2                      /              2         |                                                                                                                                                                                                                                                                                                                                                                                     
|       |      |                                                                                                                                                                                                                                                                                                                     /              y                2     /              y             |        /              y                2     /              y          |   /           /     2\            /              y                2     /              y                     /              y                2     /              y                                          |         /              2                      /              2         |                                 |        /              y                2     /              y          |   /           /     2\                                 |         /              2                      /              2         |                                                                                     |         /              2                      /              2         |                                                                                     |         /              2                      /              2         |                                       /              y           |        /              y                2     /              y          |           /     2\                                                       |         /              2                      /              2         |                                                             /              y           |        /              y                2     /              y          |           /     2\                                                                                                                                                                                                                                                                                                                                                                  
|       |      |                                                                                                                                                                                                                                                                                                             - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------    |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*\/  polar_lift\1 - x /    - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------    - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------                                 |        /              y                2     /              y          |                                 |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*\/  polar_lift\1 - x /                                 |        /              y                2     /              y          |                                                                                     |        /              y                2     /              y          |                                                                                     |        /              y                2     /              y          |                                      /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*polar_lift\1 - x /                                                       |        /              y                2     /              y          |                                                            /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*polar_lift\1 - x /                                                                                                                                                                                                                                                                                                                                                                  
|       |      |                                                                                                                                                                                                                                                                                                                   /                  /     2\           /                  /     2\    |      /                  /     2\           /                  /     2\ |                                 /                  /     2\           /                  /     2\          /                  /     2\           /                  /     2\                                 |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                 |      /                  /     2\           /                  /     2\ |                                                        |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                                                                     |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                                                                     |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                     /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                          |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |                                                           /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                                                                                                                                                                                                                                                                                                                                     
|       |      |                                                                                                                                                                                                                                                                                                                 \/         polar_lift\1 - x /         \/         polar_lift\1 - x /    \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                               \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /                                 |      /                  /     2\           /                  /     2\ |                                 \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                        |      /                  /     2\           /                  /     2\ |                                                                                     |      /                  /     2\           /                  /     2\ |                                                                                     |      /                  /     2\           /                  /     2\ |                                   \/         polar_lift\1 - x /  \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                          |      /                  /     2\           /                  /     2\ |                                                         \/         polar_lift\1 - x /  \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                     
|       |      \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                   \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                     \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                     \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                      \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
|       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
|      /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
|     ________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
|    /      2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
\  \/  1 - x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
$$\begin{cases} \int\limits_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} \begin{cases} - \frac{4 i x^{2} y \left(1 - x^{2}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{2 \pi x^{2} y \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{4 i x^{2} y \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{4 i x^{2} \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} - \frac{4 i x^{2} \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} + \frac{2 \pi x^{2} \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{4 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} - \frac{4 i y^{3} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{12 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} + \frac{4 i y \left(1 - x^{2}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 \pi y \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 i y \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} + \frac{4 i \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} + \frac{4 i \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{2 \pi \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} + \frac{4 i \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} & \text{for}\: \left|{\frac{y^{2}}{x^{2} - 1}}\right| > 1 \\- \frac{2 x^{2} y \left(1 - x^{2}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{2 x^{2} y \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} + \frac{2 x^{2} \left(1 - x^{2}\right)}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} + \frac{2 x^{2} \left(1 - x^{2}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} + \frac{2 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} + \frac{2 y^{3} \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} + \frac{6 y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} + \frac{2 y \left(1 - x^{2}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} - \frac{2 \left(1 - x^{2}\right)}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} - \frac{2 \left(1 - x^{2}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} - \frac{2 \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dy & \text{for}\: \sqrt{1 - x^{2}} > - \sqrt{1 - x^{2}} \\- \int\limits_{\sqrt{1 - x^{2}}}^{- \sqrt{1 - x^{2}}} \begin{cases} - \frac{4 i x^{2} y \left(1 - x^{2}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{2 \pi x^{2} y \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{4 i x^{2} y \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{4 i x^{2} \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} - \frac{4 i x^{2} \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} + \frac{2 \pi x^{2} \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{4 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} - \frac{4 i y^{3} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{12 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} + \frac{4 i y \left(1 - x^{2}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 \pi y \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 i y \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} + \frac{4 i \left(1 - x^{2}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} + \frac{4 i \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} - \frac{2 \pi \left(1 - x^{2}\right) \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \left(- \frac{4 x^{2} y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{4 y}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right)}{\left(4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}\right)^{2}} + \frac{4 i \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{4 x^{2} \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1} - 4 \sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - 1}} & \text{for}\: \left|{\frac{y^{2}}{x^{2} - 1}}\right| > 1 \\- \frac{2 x^{2} y \left(1 - x^{2}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + \frac{2 x^{2} y \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} + \frac{2 x^{2} \left(1 - x^{2}\right)}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} + \frac{2 x^{2} \left(1 - x^{2}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} + \frac{2 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} + \frac{2 y^{3} \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} + \frac{6 y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} + \frac{2 y \left(1 - x^{2}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} - \frac{2 \left(1 - x^{2}\right)}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} - \frac{2 \left(1 - x^{2}\right) \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \left(\frac{2 x^{2} y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}} - \frac{2 y}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - x^{2} \right)}}\right) \operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}} \right)}}{\left(2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}\right)^{2}} - \frac{2 \sqrt{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}}}{2 x^{2} \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1} - 2 \sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(1 - x^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dy & \text{otherwise} \end{cases}$$
Piecewise((Integral(Piecewise((4*i*sqrt(polar_lift(1 - x^2))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))) - 12*i*y^2*sqrt(polar_lift(1 - x^2))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))) - 4*i*x^2*sqrt(polar_lift(1 - x^2))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))) + 4*i*(1 - x^2)/((-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*sqrt(polar_lift(1 - x^2))) - 4*i*x^2*(1 - x^2)/((-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*sqrt(polar_lift(1 - x^2))) - 4*i*y^3*sqrt(polar_lift(1 - x^2))*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 - 2*pi*sqrt(-1 + y^2/polar_lift(1 - x^2))*(1 - x^2)*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 + 4*i*y*sqrt(polar_lift(1 - x^2))*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 - 4*i*y*x^2*sqrt(polar_lift(1 - x^2))*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 - 2*pi*y*(1 - x^2)/(sqrt(-1 + y^2/polar_lift(1 - x^2))*(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)) + 2*pi*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))*(1 - x^2)*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 + 4*i*sqrt(-1 + y^2/polar_lift(1 - x^2))*(1 - x^2)*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))*acosh(y/sqrt(polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 - 4*i*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))*(1 - x^2)*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))*acosh(y/sqrt(polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 + 2*pi*y*x^2*(1 - x^2)/(sqrt(-1 + y^2/polar_lift(1 - x^2))*(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)) + 4*i*y*(1 - x^2)*acosh(y/sqrt(polar_lift(1 - x^2)))/(sqrt(-1 + y^2/polar_lift(1 - x^2))*(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)) - 4*i*y*x^2*(1 - x^2)*acosh(y/sqrt(polar_lift(1 - x^2)))/(sqrt(-1 + y^2/polar_lift(1 - x^2))*(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)), Abs(y^2/(-1 + x^2)) > 1), (-2*sqrt(polar_lift(1 - x^2))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2))) - 2*(1 - x^2)/((-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))*sqrt(polar_lift(1 - x^2))) + 2*x^2*sqrt(polar_lift(1 - x^2))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2))) + 6*y^2*sqrt(polar_lift(1 - x^2))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2))) - 2*y*sqrt(polar_lift(1 - x^2))*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 + 2*x^2*(1 - x^2)/((-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))*sqrt(polar_lift(1 - x^2))) + 2*y^3*sqrt(polar_lift(1 - x^2))*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 - 2*sqrt(1 - y^2/polar_lift(1 - x^2))*(1 - x^2)*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))*asin(y/sqrt(polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 + 2*y*x^2*sqrt(polar_lift(1 - x^2))*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 + 2*y*(1 - x^2)*asin(y/sqrt(polar_lift(1 - x^2)))/(sqrt(1 - y^2/polar_lift(1 - x^2))*(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2))*(1 - x^2)*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))*asin(y/sqrt(polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 - 2*y*x^2*(1 - x^2)*asin(y/sqrt(polar_lift(1 - x^2)))/(sqrt(1 - y^2/polar_lift(1 - x^2))*(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)), True)), (y, -sqrt(1 - x^2), sqrt(1 - x^2))), sqrt(1 - x^2) > -sqrt(1 - x^2)), (-Integral(Piecewise((4*i*sqrt(polar_lift(1 - x^2))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))) - 12*i*y^2*sqrt(polar_lift(1 - x^2))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))) - 4*i*x^2*sqrt(polar_lift(1 - x^2))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))) + 4*i*(1 - x^2)/((-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*sqrt(polar_lift(1 - x^2))) - 4*i*x^2*(1 - x^2)/((-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*sqrt(polar_lift(1 - x^2))) - 4*i*y^3*sqrt(polar_lift(1 - x^2))*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 - 2*pi*sqrt(-1 + y^2/polar_lift(1 - x^2))*(1 - x^2)*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 + 4*i*y*sqrt(polar_lift(1 - x^2))*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 - 4*i*y*x^2*sqrt(polar_lift(1 - x^2))*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 - 2*pi*y*(1 - x^2)/(sqrt(-1 + y^2/polar_lift(1 - x^2))*(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)) + 2*pi*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))*(1 - x^2)*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 + 4*i*sqrt(-1 + y^2/polar_lift(1 - x^2))*(1 - x^2)*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))*acosh(y/sqrt(polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 - 4*i*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))*(1 - x^2)*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))*acosh(y/sqrt(polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 + 2*pi*y*x^2*(1 - x^2)/(sqrt(-1 + y^2/polar_lift(1 - x^2))*(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)) + 4*i*y*(1 - x^2)*acosh(y/sqrt(polar_lift(1 - x^2)))/(sqrt(-1 + y^2/polar_lift(1 - x^2))*(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)) - 4*i*y*x^2*(1 - x^2)*acosh(y/sqrt(polar_lift(1 - x^2)))/(sqrt(-1 + y^2/polar_lift(1 - x^2))*(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)), Abs(y^2/(-1 + x^2)) > 1), (-2*sqrt(polar_lift(1 - x^2))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2))) - 2*(1 - x^2)/((-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))*sqrt(polar_lift(1 - x^2))) + 2*x^2*sqrt(polar_lift(1 - x^2))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2))) + 6*y^2*sqrt(polar_lift(1 - x^2))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2))) - 2*y*sqrt(polar_lift(1 - x^2))*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 + 2*x^2*(1 - x^2)/((-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))*sqrt(polar_lift(1 - x^2))) + 2*y^3*sqrt(polar_lift(1 - x^2))*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 - 2*sqrt(1 - y^2/polar_lift(1 - x^2))*(1 - x^2)*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))*asin(y/sqrt(polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 + 2*y*x^2*sqrt(polar_lift(1 - x^2))*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 + 2*y*(1 - x^2)*asin(y/sqrt(polar_lift(1 - x^2)))/(sqrt(1 - y^2/polar_lift(1 - x^2))*(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2))*(1 - x^2)*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))*asin(y/sqrt(polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 - 2*y*x^2*(1 - x^2)*asin(y/sqrt(polar_lift(1 - x^2)))/(sqrt(1 - y^2/polar_lift(1 - x^2))*(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)), True)), (y, sqrt(1 - x^2), -sqrt(1 - x^2))), True))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.