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Límite de la función:
sqrt(1-x^2-y^2)
Forma canónica:
sqrt(1-x^2-y^2)
Expresiones idénticas
sqrt(uno -x^ dos -y^ dos)
raíz cuadrada de (1 menos x al cuadrado menos y al cuadrado )
raíz cuadrada de (uno menos x en el grado dos menos y en el grado dos)
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sqrt(1-x2-y2)
sqrt1-x2-y2
sqrt(1-x²-y²)
sqrt(1-x en el grado 2-y en el grado 2)
sqrt1-x^2-y^2
sqrt(1-x^2-y^2)dx
Expresiones semejantes
sqrt(1-x^2+y^2)
sqrt(1+x^2-y^2)
Expresiones con funciones
Raíz cuadrada sqrt
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sqrt(x^2+6*x+18)/(x+3)
sqrt(-x^2+4)
sqrt(x-2)/(1+sqrt(x-2))
sqrt((x^2)-1)/sqrt((x^6)-3)

Resolución de integrales/ 1-x^2/ 2-y^2/ x^2-y/ sqrt(1-x^2-y^2)

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

Solución

Ha introducido [src]

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0

\int\limits_{0}^{1} \sqrt{- y^{2} + \left(1 - x^{2}\right)}\, dx

Integral(sqrt(1 - x^2 - y^2), (x, 0, 1))

Respuesta (Indefinida) [src]

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 |   /      2    2           ||      \/          polar_lift\1 - y /         \/          polar_lift\1 - y /        \/          polar_lift\1 - y /         \/          polar_lift\1 - y /        \/          polar_lift\1 - y /         \/          polar_lift\1 - y /        \/          polar_lift\1 - y /         \/          polar_lift\1 - y /        \/          polar_lift\1 - y /         \/          polar_lift\1 - y /         \/          polar_lift\1 - y /         \/          polar_lift\1 - y /          \/          polar_lift\1 - y /         \/          polar_lift\1 - y /                      |
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                             ||                                                                                                            x *\/  polar_lift\1 - y /                                                  x*\/  polar_lift\1 - y /                                                  x*y *\/  polar_lift\1 - y /                                                                       \\/  polar_lift\1 - y / /                                                    \\/  polar_lift\1 - y / /                                                                                                         |
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                             \\                                                                                         \/         polar_lift\1 - y /         \/         polar_lift\1 - y /        \/         polar_lift\1 - y /         \/         polar_lift\1 - y /        \/         polar_lift\1 - y /         \/         polar_lift\1 - y /        \/         polar_lift\1 - y /         \/         polar_lift\1 - y /        \/         polar_lift\1 - y /         \/         polar_lift\1 - y /                                                                                                          /

\int \sqrt{- y^{2} + \left(1 - x^{2}\right)}\, dx = C + \begin{cases} - \frac{2 i x^{3} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} + \frac{2 i x \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} - \frac{2 i y^{2} \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} + \frac{\pi y^{2} \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} + \frac{2 i \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} - \frac{\pi \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2} - 1}}\right| > 1 \\\frac{x^{3} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} + \frac{x y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{x \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} + \frac{y^{2} \left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{\left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}

Simplificar

Respuesta [src]

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 |  |                                                                                                                                                                                                                                                                                                                                              pi*   /   -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|   2*I*x *\/  polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------|                                                                                                          2*I*x*\/  polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------|   pi*y *   /   -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|                                                                                                                                     2*I*x*y *\/  polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------|   2*I*   /   -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                     2*I*y *   /   -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                                                                                                                                                                          
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 |  |                              ____________________                                                           ____________________                                                         ____________________                                                                                                                                                                            |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                  |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                                                                                                                        |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                                  |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                                                                                                                                                      |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                                |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                                                                                                                                                                                                    |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                                                                               |   ____________________|                                                                                                           |   ____________________|                                                             
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 |  |                       2*I*\/  polar_lift\1 - y /                                                  6*I*x *\/  polar_lift\1 - y /                                                2*I*y *\/  polar_lift\1 - y /                                                                      2*I*\1 - y /                                                                                           \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                  \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                             2*I*y *\1 - y /                                                                            \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                  \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                            pi*x*\1 - y /                                                                                             \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                                                          pi*x*y *\1 - y /                                                                                                          \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                                                               \\/  polar_lift\1 - y / /                                                                                                           \\/  polar_lift\1 - y / /                                                 x           
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 |  |        /               x                2     /               x                     /               x                2     /               x                     /               x                2     /               x             |        /               x                2     /               x          |   /           /     2\                                        |         /               2                      /               2         |                                                                     |         /               2                      /               2         |                                  |        /               x                2     /               x          |   /           /     2\                                 |         /               2                      /               2         |                                                                      |         /               2                      /               2         |                                              /               x           |        /               x                2     /               x          |           /     2\                                 |         /               2                      /               2         |                                                                                       |         /               2                      /               2         |                                                            /               x           |        /               x                2     /               x          |           /     2\                                                        |         /               2                      /               2         |                                                              /               x           |        /               x                2     /               x          |           /     2\       /               x           |        /               x                2     /               x          |           /     2\                      
 |  |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*\/  polar_lift\1 - y /                                        |        /               x                2     /               x          |                                                                     |        /               x                2     /               x          |                                  |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*\/  polar_lift\1 - y /                                 |        /               x                2     /               x          |                                                                      |        /               x                2     /               x          |                                             /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\1 - y /                                 |        /               x                2     /               x          |                                                                                       |        /               x                2     /               x          |                                                           /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\1 - y /                                                        |        /               x                2     /               x          |                                                             /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\1 - y /      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\1 - y /                      
 |  |      /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\    |      /                   /     2\           /                   /     2\ |                                                               |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                     |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                  |      /                   /     2\           /                   /     2\ |                                                        |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                      |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                            /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                    |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                                       |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                          /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                                           |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                            /                   /     2\  |      /                   /     2\           /                   /     2\ |                        /                   /     2\  |      /                   /     2\           /                   /     2\ |                                         
 |  |    \/          polar_lift\1 - y /         \/          polar_lift\1 - y /        \/          polar_lift\1 - y /         \/          polar_lift\1 - y /        \/          polar_lift\1 - y /         \/          polar_lift\1 - y /    \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                               |      /                   /     2\           /                   /     2\ |                                                                     |      /                   /     2\           /                   /     2\ |                                  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                        |      /                   /     2\           /                   /     2\ |                                                                      |      /                   /     2\           /                   /     2\ |                                          \/          polar_lift\1 - y /  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                    |      /                   /     2\           /                   /     2\ |                                                                                       |      /                   /     2\           /                   /     2\ |                                                        \/          polar_lift\1 - y /  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                           |      /                   /     2\           /                   /     2\ |                                                          \/          polar_lift\1 - y /  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                      \/          polar_lift\1 - y /  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                         
 |  |                                                                                                                                                                                                                                                                                                                                                                                  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                     \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                                                                                                      \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                      \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                                                                                                                                          \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                       \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                                                                                                                                                                               \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                                                                                                                                                                                                                                                                                 
 |  <                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      dx
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 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       3   /           /     2\  |                         2*x                                                2*x*y                      |                                                                                                            /           /     2\  |                         2*x                                                2*x*y                      |      2   /           /     2\  |                         2*x                                                2*x*y                      |       /              x           /     2\ |                         2*x                                                2*x*y                      |     /           x           \                                                                                                                                   2     /              x           /     2\ |                         2*x                                                2*x*y                      |     /           x           \                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      x *\/  polar_lift\1 - y / *|- ------------------------------------------------- + -------------------------------------------------|                                                                                                        x*\/  polar_lift\1 - y / *|- ------------------------------------------------- + -------------------------------------------------|   x*y *\/  polar_lift\1 - y / *|- ------------------------------------------------- + -------------------------------------------------|      /   1 - ------------------ *\1 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|                                                                                                                                  y *   /   1 - ------------------ *\1 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
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 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |       /              2                                    /              2                            |                                                                                                                                  |       /              2                                    /              2                            |                                |       /              2                                    /              2                            |   \/         polar_lift\1 - y /           |       /              2                                    /              2                            |     |  /           /     2\ |                                               /     2\     /           x           \                                                \/         polar_lift\1 - y /           |       /              2                                    /              2                            |     |  /           /     2\ |                                              2 /     2\     /           x           \                                                                                                                                                                                                                                                                                                                                                                              
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |      /              x                     /     2\       /              x                     /     2\|                                                                                                                                  |      /              x                     /     2\       /              x                     /     2\|                                |      /              x                     /     2\       /              x                     /     2\|                                           |      /              x                     /     2\       /              x                     /     2\|     \\/  polar_lift\1 - y / /                                             x*\1 - y /*asin|-----------------------|                                                                                        |      /              x                     /     2\       /              x                     /     2\|     \\/  polar_lift\1 - y / /                                           x*y *\1 - y /*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                              
 |  |                                                                                                                                                                                                                                                                                                                                           ____________________                                                         ____________________                                                                                                                                                             ____________________                                                    |     /   1 - ------------------ *polar_lift\1 - y /      /   1 - ------------------ *polar_lift\1 - y /|                                                                                                                                  |     /   1 - ------------------ *polar_lift\1 - y /      /   1 - ------------------ *polar_lift\1 - y /|                                |     /   1 - ------------------ *polar_lift\1 - y /      /   1 - ------------------ *polar_lift\1 - y /|                                           |     /   1 - ------------------ *polar_lift\1 - y /      /   1 - ------------------ *polar_lift\1 - y /|                                                                                          |   ____________________|                                                                                        |     /   1 - ------------------ *polar_lift\1 - y /      /   1 - ------------------ *polar_lift\1 - y /|                                                                                           |   ____________________|                                                                                                                                                                                                                                                                                                                                                                              
 |  |                                                                                                                                                                                                                                                                                                                                          /           /     2\                                                     2   /           /     2\                                                                              2                                                                          2   /           /     2\                                                     |    /                  /     2\                         /                  /     2\                    |                                               2 /     2\                                                                         |    /                  /     2\                         /                  /     2\                    |                                |    /                  /     2\                         /                  /     2\                    |                                           |    /                  /     2\                         /                  /     2\                    |                                                                                          |  /           /     2\ |                                                                                        |    /                  /     2\                         /                  /     2\                    |                                                                                           |  /           /     2\ |                                                                                                                                                                                                                                                                                                                                                                              
 |  |                                                                                                                                                                                                                                                                                                                                        \/  polar_lift\1 - y /                                                    y *\/  polar_lift\1 - y /                                                                         1 - y                                                                        3*x *\/  polar_lift\1 - y /                                                     \  \/         polar_lift\1 - y /                       \/         polar_lift\1 - y /                    /                                              y *\1 - y /                                                                         \  \/         polar_lift\1 - y /                       \/         polar_lift\1 - y /                    /                                \  \/         polar_lift\1 - y /                       \/         polar_lift\1 - y /                    /                                           \  \/         polar_lift\1 - y /                       \/         polar_lift\1 - y /                    /                                                                                          \\/  polar_lift\1 - y / /                                                                                        \  \/         polar_lift\1 - y /                       \/         polar_lift\1 - y /                    /                                                                                           \\/  polar_lift\1 - y / /                                                                                                                                                                                                                                                                                                                                                                              
 |  |                                                                                                                                                                                                                                                                                                              - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------ + -------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------                                                                                                                                                                                                                                                                                                                     otherwise       
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 |  |                                                                                                                                                                                                                                                                                                                         /              2                      /              2                     /              2                      /              2            |         /              2                      /              2         |    ____________________            /              2                      /              2                                        /          ________________________              ________________________\                                 |         /              2                      /              2         |    ____________________                               /          ________________________              ________________________\                                                             /          ________________________              ________________________\                                                                                    /          ________________________              ________________________\                                                           /              2          |         /              2                      /              2         |                                                                         /          ________________________              ________________________\                                                             /              2          |         /              2                      /              2         |                                                                                                                                                                                                                                                                                                                                                        
 |  |                                                                                                                                                                                                                                                                                                                        /              x                2     /              x                     /              x                2     /              x             |        /              x                2     /              x          |   /           /     2\            /              x                2     /              x                                         |         /              2                      /              2         |                                 |        /              x                2     /              x          |   /           /     2\                                |         /              2                      /              2         |                                                             |         /              2                      /              2         |                                                                                    |         /              2                      /              2         |                                                          /              x           |        /              x                2     /              x          |           /     2\                                                      |         /              2                      /              2         |                                                            /              x           |        /              x                2     /              x          |           /     2\                                                                                                                                                                                                                                                                                                                                     
 |  |                                                                                                                                                                                                                                                                                                                - 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------    - 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------    |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*\/  polar_lift\1 - y /    - 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------                                |        /              x                2     /              x          |                                 |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*\/  polar_lift\1 - y /                                |        /              x                2     /              x          |                                                             |        /              x                2     /              x          |                                                                                    |        /              x                2     /              x          |                                                         /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*polar_lift\1 - y /                                                      |        /              x                2     /              x          |                                                           /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*polar_lift\1 - y /                                                                                                                                                                                                                                                                                                                                     
 |  |                                                                                                                                                                                                                                                                                                                      /                  /     2\           /                  /     2\          /                  /     2\           /                  /     2\    |      /                  /     2\           /                  /     2\ |                                 /                  /     2\           /                  /     2\                                |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                 |      /                  /     2\           /                  /     2\ |                                                       |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                             |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                                                    |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                        /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                         |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                          /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                                                                                                                                                                                                                                                                                                        
 |  |                                                                                                                                                                                                                                                                                                                    \/         polar_lift\1 - y /         \/         polar_lift\1 - y /        \/         polar_lift\1 - y /         \/         polar_lift\1 - y /    \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                               \/         polar_lift\1 - y /         \/         polar_lift\1 - y /                                |      /                  /     2\           /                  /     2\ |                                 \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                       |      /                  /     2\           /                  /     2\ |                                                             |      /                  /     2\           /                  /     2\ |                                                                                    |      /                  /     2\           /                  /     2\ |                                                      \/         polar_lift\1 - y /  \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                         |      /                  /     2\           /                  /     2\ |                                                        \/         polar_lift\1 - y /  \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                                                                                                                                                                                                        
 |  \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                  \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                             \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                    \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                                                                                        \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         
 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           
/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
0

\int\limits_{0}^{1} \begin{cases} - \frac{2 i x^{3} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \left(1 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{\pi x y^{2} \left(1 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{2 i x \left(1 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{\pi x \left(1 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{2 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{\pi y^{2} \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(1 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} - \frac{2 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} + \frac{2 i \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{\pi \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{2 i \left(1 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} + \frac{2 i \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{y^{2} - 1}\right|} > 1 \\\frac{x^{3} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{3 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{x y^{2} \left(1 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{x y^{2} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{x \left(1 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(1 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} + \frac{y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{\left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} - \frac{1 - y^{2}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} - \frac{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dx

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 |  |                                                                                                                                                                                                                                                                                                                                              pi*   /   -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|   2*I*x *\/  polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------|                                                                                                          2*I*x*\/  polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------|   pi*y *   /   -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|                                                                                                                                     2*I*x*y *\/  polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------|   2*I*   /   -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                     2*I*y *   /   -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                                                                                                                                                                          
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 |  |                       2*I*\/  polar_lift\1 - y /                                                  6*I*x *\/  polar_lift\1 - y /                                                2*I*y *\/  polar_lift\1 - y /                                                                      2*I*\1 - y /                                                                                           \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                  \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                             2*I*y *\1 - y /                                                                            \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                  \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                            pi*x*\1 - y /                                                                                             \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                                                          pi*x*y *\1 - y /                                                                                                          \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                                                               \\/  polar_lift\1 - y / /                                                                                                           \\/  polar_lift\1 - y / /                                                 x           
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 |  |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*\/  polar_lift\1 - y /                                        |        /               x                2     /               x          |                                                                     |        /               x                2     /               x          |                                  |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*\/  polar_lift\1 - y /                                 |        /               x                2     /               x          |                                                                      |        /               x                2     /               x          |                                             /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\1 - y /                                 |        /               x                2     /               x          |                                                                                       |        /               x                2     /               x          |                                                           /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\1 - y /                                                        |        /               x                2     /               x          |                                                             /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\1 - y /      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\1 - y /                      
 |  |      /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\    |      /                   /     2\           /                   /     2\ |                                                               |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                     |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                  |      /                   /     2\           /                   /     2\ |                                                        |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                      |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                            /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                    |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                                       |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                          /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                                           |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                            /                   /     2\  |      /                   /     2\           /                   /     2\ |                        /                   /     2\  |      /                   /     2\           /                   /     2\ |                                         
 |  |    \/          polar_lift\1 - y /         \/          polar_lift\1 - y /        \/          polar_lift\1 - y /         \/          polar_lift\1 - y /        \/          polar_lift\1 - y /         \/          polar_lift\1 - y /    \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                               |      /                   /     2\           /                   /     2\ |                                                                     |      /                   /     2\           /                   /     2\ |                                  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                        |      /                   /     2\           /                   /     2\ |                                                                      |      /                   /     2\           /                   /     2\ |                                          \/          polar_lift\1 - y /  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                    |      /                   /     2\           /                   /     2\ |                                                                                       |      /                   /     2\           /                   /     2\ |                                                        \/          polar_lift\1 - y /  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                           |      /                   /     2\           /                   /     2\ |                                                          \/          polar_lift\1 - y /  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                      \/          polar_lift\1 - y /  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                         
 |  |                                                                                                                                                                                                                                                                                                                                                                                  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                     \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                                                                                                      \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                      \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                                                                                                                                          \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                       \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                                                                                                                                                                               \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                                                                                                                                                                                                                                                                                 
 |  <                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      dx
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       ________________________                                                                                                                                                                                                                                                                                            ________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
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 |  |                                                                                                                                                                                                                                                                                                                    \/         polar_lift\1 - y /         \/         polar_lift\1 - y /        \/         polar_lift\1 - y /         \/         polar_lift\1 - y /    \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                               \/         polar_lift\1 - y /         \/         polar_lift\1 - y /                                |      /                  /     2\           /                  /     2\ |                                 \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                       |      /                  /     2\           /                  /     2\ |                                                             |      /                  /     2\           /                  /     2\ |                                                                                    |      /                  /     2\           /                  /     2\ |                                                      \/         polar_lift\1 - y /  \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                         |      /                  /     2\           /                  /     2\ |                                                        \/         polar_lift\1 - y /  \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                                                                                                                                                                                                        
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0

\int\limits_{0}^{1} \begin{cases} - \frac{2 i x^{3} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \left(1 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{\pi x y^{2} \left(1 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{2 i x \left(1 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{\pi x \left(1 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{2 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{\pi y^{2} \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(1 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} - \frac{2 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} + \frac{2 i \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{\pi \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{2 i \left(1 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} + \frac{2 i \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{y^{2} - 1}\right|} > 1 \\\frac{x^{3} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{3 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{x y^{2} \left(1 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{x y^{2} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{x \left(1 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(1 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} + \frac{y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{\left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} - \frac{1 - y^{2}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} - \frac{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dx

Integral(Piecewise((2*i*sqrt(polar_lift(1 - y^2))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2))) - 6*i*x^2*sqrt(polar_lift(1 - y^2))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2))) - 2*i*y^2*sqrt(polar_lift(1 - y^2))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2))) + 2*i*(1 - y^2)/((-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))*sqrt(polar_lift(1 - y^2))) - pi*sqrt(-1 + x^2/polar_lift(1 - y^2))*(1 - y^2)*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 - 2*i*x^3*sqrt(polar_lift(1 - y^2))*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 - 2*i*y^2*(1 - y^2)/((-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))*sqrt(polar_lift(1 - y^2))) + 2*i*x*sqrt(polar_lift(1 - y^2))*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 + pi*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2))*(1 - y^2)*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 - pi*x*(1 - y^2)/(sqrt(-1 + x^2/polar_lift(1 - y^2))*(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))*polar_lift(1 - y^2)) - 2*i*x*y^2*sqrt(polar_lift(1 - y^2))*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 + 2*i*sqrt(-1 + x^2/polar_lift(1 - y^2))*(1 - y^2)*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))*acosh(x/sqrt(polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 + pi*x*y^2*(1 - y^2)/(sqrt(-1 + x^2/polar_lift(1 - y^2))*(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))*polar_lift(1 - y^2)) - 2*i*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2))*(1 - y^2)*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))*acosh(x/sqrt(polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 + 2*i*x*(1 - y^2)*acosh(x/sqrt(polar_lift(1 - y^2)))/(sqrt(-1 + x^2/polar_lift(1 - y^2))*(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))*polar_lift(1 - y^2)) - 2*i*x*y^2*(1 - y^2)*acosh(x/sqrt(polar_lift(1 - y^2)))/(sqrt(-1 + x^2/polar_lift(1 - y^2))*(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))*polar_lift(1 - y^2)), x^2/|-1 + y^2| > 1), (-sqrt(polar_lift(1 - y^2))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2))) + y^2*sqrt(polar_lift(1 - y^2))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2))) - (1 - y^2)/((-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))*sqrt(polar_lift(1 - y^2))) + 3*x^2*sqrt(polar_lift(1 - y^2))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2))) + x^3*sqrt(polar_lift(1 - y^2))*(-2*x/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))^2 + y^2*(1 - y^2)/((-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))*sqrt(polar_lift(1 - y^2))) - x*sqrt(polar_lift(1 - y^2))*(-2*x/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))^2 + x*y^2*sqrt(polar_lift(1 - y^2))*(-2*x/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))^2 - sqrt(1 - x^2/polar_lift(1 - y^2))*(1 - y^2)*(-2*x/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))*asin(x/sqrt(polar_lift(1 - y^2)))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))^2 + x*(1 - y^2)*asin(x/sqrt(polar_lift(1 - y^2)))/(sqrt(1 - x^2/polar_lift(1 - y^2))*(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))*polar_lift(1 - y^2)) + y^2*sqrt(1 - x^2/polar_lift(1 - y^2))*(1 - y^2)*(-2*x/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))*asin(x/sqrt(polar_lift(1 - y^2)))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))^2 - x*y^2*(1 - y^2)*asin(x/sqrt(polar_lift(1 - y^2)))/(sqrt(1 - x^2/polar_lift(1 - y^2))*(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))*polar_lift(1 - y^2)), True)), (x, 0, 1))

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

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

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