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Expresiones idénticas
(R^ dos -x^ dos -y^ dos)^(uno / dos)
(R al cuadrado menos x al cuadrado menos y al cuadrado ) en el grado (1 dividir por 2)
(R en el grado dos menos x en el grado dos menos y en el grado dos) en el grado (uno dividir por dos)
(R2-x2-y2)(1/2)
R2-x2-y21/2
(R²-x²-y²)^(1/2)
(R en el grado 2-x en el grado 2-y en el grado 2) en el grado (1/2)
R^2-x^2-y^2^1/2
(R^2-x^2-y^2)^(1 dividir por 2)
(R^2-x^2-y^2)^(1/2)dx
Expresiones semejantes
(R^2+x^2-y^2)^(1/2)
(R^2-x^2+y^2)^(1/2)

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

Integral de (R^2-x^2-y^2)^(1/2) dx

Solución

Ha introducido [src]

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0

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

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

Respuesta (Indefinida) [src]

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 |   /  2    2    2           ||         \/          polar_lift\r  - y /         \/          polar_lift\r  - y /           \/          polar_lift\r  - y /         \/          polar_lift\r  - y /           \/          polar_lift\r  - y /         \/          polar_lift\r  - y /           \/          polar_lift\r  - y /         \/          polar_lift\r  - y /           \/          polar_lift\r  - y /         \/          polar_lift\r  - y /            \/          polar_lift\r  - y /         \/          polar_lift\r  - y /             \/          polar_lift\r  - y /         \/          polar_lift\r  - y /                     |
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                              \\                                                                                                 \/         polar_lift\r  - y /         \/         polar_lift\r  - y /           \/         polar_lift\r  - y /         \/         polar_lift\r  - y /           \/         polar_lift\r  - y /         \/         polar_lift\r  - y /           \/         polar_lift\r  - y /         \/         polar_lift\r  - y /           \/         polar_lift\r  - y /         \/         polar_lift\r  - y /                                                                                                              /

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

Simplificar

Respuesta [src]

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 |  |       2     /                x                2     /                x                  2     /                x                2     /                x                  2     /                x                2     /                x                                            |            /                2                      /                2         |                                  |     2     /                x                2     /                x          |   /           / 2    2\    |     2     /                x                2     /                x          |   /           / 2    2\                                           |            /                2                      /                2         |                                                                                  |            /                2                      /                2         |                                                                           |            /                2                      /                2         |                                                                   |            /                2                      /                2         |                                       /                x           |     2     /                x                2     /                x          |           / 2    2\       /                x           |     2     /                x                2     /                x          |           / 2    2\                                                           |            /                2                      /                2         |                                                                                                                   |            /                2                      /                2         |                                                               /                x           |     2     /                x                2     /                x          |           / 2    2\       /                x           |     2     /                x                2     /                x          |           / 2    2\                      
 |  |  - 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + -------------------    - 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + -------------------    - 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + -------------------                                   |     2     /                x                2     /                x          |                                  |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |*\/  polar_lift\r  - y /    |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |*\/  polar_lift\r  - y /                                           |     2     /                x                2     /                x          |                                                                                  |     2     /                x                2     /                x          |                                                                           |     2     /                x                2     /                x          |                                                                   |     2     /                x                2     /                x          |                                      /   -1 + ------------------- *|- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |*polar_lift\r  - y /      /   -1 + ------------------- *|- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |*polar_lift\r  - y /                                                           |     2     /                x                2     /                x          |                                                                                                                   |     2     /                x                2     /                x          |                                                              /   -1 + ------------------- *|- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |*polar_lift\r  - y /      /   -1 + ------------------- *|- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |*polar_lift\r  - y /                      
 |  |           /                   / 2    2\           /                   / 2    2\             /                   / 2    2\           /                   / 2    2\             /                   / 2    2\           /                   / 2    2\                                   |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                  |         /                   / 2    2\           /                   / 2    2\ |                            |         /                   / 2    2\           /                   / 2    2\ |                                                                   |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                                                                  |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                                                           |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                                                   |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                     /                   / 2    2\  |         /                   / 2    2\           /                   / 2    2\ |                         /                   / 2    2\  |         /                   / 2    2\           /                   / 2    2\ |                                                                               |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                                                                                                   |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                                             /                   / 2    2\  |         /                   / 2    2\           /                   / 2    2\ |                         /                   / 2    2\  |         /                   / 2    2\           /                   / 2    2\ |                                          
 |  |         \/          polar_lift\r  - y /         \/          polar_lift\r  - y /           \/          polar_lift\r  - y /         \/          polar_lift\r  - y /           \/          polar_lift\r  - y /         \/          polar_lift\r  - y /                                   |         /                   / 2    2\           /                   / 2    2\ |                                  \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                            \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                   |         /                   / 2    2\           /                   / 2    2\ |                                                                                  |         /                   / 2    2\           /                   / 2    2\ |                                                                           |         /                   / 2    2\           /                   / 2    2\ |                                                                   |         /                   / 2    2\           /                   / 2    2\ |                                   \/          polar_lift\r  - y /  \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                       \/          polar_lift\r  - y /  \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                               |         /                   / 2    2\           /                   / 2    2\ |                                                                                                                   |         /                   / 2    2\           /                   / 2    2\ |                                                           \/          polar_lift\r  - y /  \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                       \/          polar_lift\r  - y /  \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                          
 |  |                                                                                                                                                                                                                                                                                       \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                                                                                                                                                                                                                                                   \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                                  \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                           \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                   \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                                                                                                                                                                                                                                                                                                                             \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                                                                   \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                                                                                                                                                                                                                                                                                                                
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 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             _________________________                                                                                                                                                                   _________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            _____________________ /                              2                                                     2                      \                                                                                                                                                                                                                                 _____________________ /                              2                                                     2                      \           _____________________ /                              2                                                     2                      \           /               2                    /                              2                                                     2                      \                                          /               2                    /                              2                                                     2                      \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       3   /           / 2    2\  |                         2*x*r                                                 2*x*y                       |                                                                                                                                                                                                                            2   /           / 2    2\  |                         2*x*r                                                 2*x*y                       |      2   /           / 2    2\  |                         2*x*r                                                 2*x*y                       |    2     /               x           / 2    2\ |                         2*x*r                                                 2*x*y                       |     /           x            \    2     /               x           / 2    2\ |                         2*x*r                                                 2*x*y                       |     /           x            \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      x *\/  polar_lift\r  - y / *|- --------------------------------------------------- + ---------------------------------------------------|                                                                                                                                                                                                                         x*y *\/  polar_lift\r  - y / *|- --------------------------------------------------- + ---------------------------------------------------|   x*r *\/  polar_lift\r  - y / *|- --------------------------------------------------- + ---------------------------------------------------|   y *   /   1 - ------------------- *\r  - y /*|- --------------------------------------------------- + ---------------------------------------------------|*asin|------------------------|   r *   /   1 - ------------------- *\r  - y /*|- --------------------------------------------------- + ---------------------------------------------------|*asin|------------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |        _________________________                             _________________________                    |                                                                                                                                                                                                                                                       |        _________________________                             _________________________                    |                                 |        _________________________                             _________________________                    |        /                  / 2    2\            |        _________________________                             _________________________                    |     |   _____________________|        /                  / 2    2\            |        _________________________                             _________________________                    |     |   _____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |       /               2                                     /               2                             |                                                                                                                                                                                                                                                       |       /               2                                     /               2                             |                                 |       /               2                                     /               2                             |      \/         polar_lift\r  - y /            |       /               2                                     /               2                             |     |  /           / 2    2\ |      \/         polar_lift\r  - y /            |       /               2                                     /               2                             |     |  /           / 2    2\ |                                                 2 / 2    2\     /           x            \                                                                                            2 / 2    2\     /           x            \                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |      /               x                     / 2    2\       /               x                     / 2    2\|                                                                                                                                                                                                                                                       |      /               x                     / 2    2\       /               x                     / 2    2\|                                 |      /               x                     / 2    2\       /               x                     / 2    2\|                                                |      /               x                     / 2    2\       /               x                     / 2    2\|     \\/  polar_lift\r  - y / /                                                |      /               x                     / 2    2\       /               x                     / 2    2\|     \\/  polar_lift\r  - y / /                                              x*r *\r  - y /*asin|------------------------|                                                                                         x*y *\r  - y /*asin|------------------------|                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                                     _____________________                                                           _____________________                                                            _____________________                                                       |     /   1 - ------------------- *polar_lift\r  - y /      /   1 - ------------------- *polar_lift\r  - y /|                                                                                                                                                                                                                                                       |     /   1 - ------------------- *polar_lift\r  - y /      /   1 - ------------------- *polar_lift\r  - y /|                                 |     /   1 - ------------------- *polar_lift\r  - y /      /   1 - ------------------- *polar_lift\r  - y /|                                                |     /   1 - ------------------- *polar_lift\r  - y /      /   1 - ------------------- *polar_lift\r  - y /|                                                                               |     /   1 - ------------------- *polar_lift\r  - y /      /   1 - ------------------- *polar_lift\r  - y /|                                                                                                |   _____________________|                                                                                                            |   _____________________|                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                                2   /           / 2    2\                                                       2   /           / 2    2\                                                        2   /           / 2    2\                                                        |    /                  / 2    2\                          /                  / 2    2\                     |                                                  2 / 2    2\                                                                                                2 / 2    2\                                                                               |    /                  / 2    2\                          /                  / 2    2\                     |                                 |    /                  / 2    2\                          /                  / 2    2\                     |                                                |    /                  / 2    2\                          /                  / 2    2\                     |                                                                               |    /                  / 2    2\                          /                  / 2    2\                     |                                                                                                |  /           / 2    2\ |                                                                                                            |  /           / 2    2\ |                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                               y *\/  polar_lift\r  - y /                                                      r *\/  polar_lift\r  - y /                                                     3*x *\/  polar_lift\r  - y /                                                        \  \/         polar_lift\r  - y /                        \/         polar_lift\r  - y /                     /                                                 y *\r  - y /                                                                                               r *\r  - y /                                                                               \  \/         polar_lift\r  - y /                        \/         polar_lift\r  - y /                     /                                 \  \/         polar_lift\r  - y /                        \/         polar_lift\r  - y /                     /                                                \  \/         polar_lift\r  - y /                        \/         polar_lift\r  - y /                     /                                                                               \  \/         polar_lift\r  - y /                        \/         polar_lift\r  - y /                     /                                                                                                \\/  polar_lift\r  - y / /                                                                                                            \\/  polar_lift\r  - y / /                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                      ----------------------------------------------------------------------------- - ----------------------------------------------------------------------------- + ----------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------------------------------------------                                                                                                                                                                                                                                                                                                                                            otherwise       
 |  |                                                                                                                                                                                                                                                                                                                                                   _________________________              _________________________                _________________________              _________________________                _________________________              _________________________                                                                                                               2                               /             _________________________              _________________________\                            /             _________________________              _________________________\                                                                                                                                         2                                                                                                                                             2                                                                                                                                                                    2                                                                                                                                                                                           2                                                             _________________________ /             _________________________              _________________________\                             _________________________ /             _________________________              _________________________\                                                                                                                                                                                                                                                                                                                                                                                
 |  |                                                                                                                                                                                                                                                                                                                                                  /               2                      /               2                        /               2                      /               2                        /               2                      /               2                                         /             _________________________              _________________________\                                |            /               2                      /               2         |    _____________________   |            /               2                      /               2         |    _____________________                                 /             _________________________              _________________________\                                                               /             _________________________              _________________________\                                                                                      /             _________________________              _________________________\                                                                                                             /             _________________________              _________________________\                                                             /               2          |            /               2                      /               2         |                            /               2          |            /               2                      /               2         |                                                                                                                                                                                                                                                                                                                                                                                
 |  |                                                                                                                                                                                                                                                                                                                                           2     /               x                2     /               x                  2     /               x                2     /               x                  2     /               x                2     /               x                                          |            /               2                      /               2         |                                |     2     /               x                2     /               x          |   /           / 2    2\    |     2     /               x                2     /               x          |   /           / 2    2\                                  |            /               2                      /               2         |                                                               |            /               2                      /               2         |                                                                                      |            /               2                      /               2         |                                                                                                             |            /               2                      /               2         |                                                            /               x           |     2     /               x                2     /               x          |           / 2    2\       /               x           |     2     /               x                2     /               x          |           / 2    2\                                                                                                                                                                                                                                                                                                                                                            
 |  |                                                                                                                                                                                                                                                                                                                                      - 2*r *   /   1 - -------------------  + 2*y *   /   1 - -------------------    - 2*r *   /   1 - -------------------  + 2*y *   /   1 - -------------------    - 2*r *   /   1 - -------------------  + 2*y *   /   1 - -------------------                                 |     2     /               x                2     /               x          |                                |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |*\/  polar_lift\r  - y /    |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |*\/  polar_lift\r  - y /                                  |     2     /               x                2     /               x          |                                                               |     2     /               x                2     /               x          |                                                                                      |     2     /               x                2     /               x          |                                                                                                             |     2     /               x                2     /               x          |                                                           /   1 - ------------------- *|- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |*polar_lift\r  - y /      /   1 - ------------------- *|- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |*polar_lift\r  - y /                                                                                                                                                                                                                                                                                                                                                            
 |  |                                                                                                                                                                                                                                                                                                                                               /                  / 2    2\           /                  / 2    2\             /                  / 2    2\           /                  / 2    2\             /                  / 2    2\           /                  / 2    2\                                 |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |                                |         /                  / 2    2\           /                  / 2    2\ |                            |         /                  / 2    2\           /                  / 2    2\ |                                                          |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |                                                               |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |                                                                                      |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |                                                                                                             |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |                                                          /                  / 2    2\  |         /                  / 2    2\           /                  / 2    2\ |                         /                  / 2    2\  |         /                  / 2    2\           /                  / 2    2\ |                                                                                                                                                                                                                                                                                                                                                                                
 |  |                                                                                                                                                                                                                                                                                                                                             \/         polar_lift\r  - y /         \/         polar_lift\r  - y /           \/         polar_lift\r  - y /         \/         polar_lift\r  - y /           \/         polar_lift\r  - y /         \/         polar_lift\r  - y /                                 |         /                  / 2    2\           /                  / 2    2\ |                                \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                            \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                          |         /                  / 2    2\           /                  / 2    2\ |                                                               |         /                  / 2    2\           /                  / 2    2\ |                                                                                      |         /                  / 2    2\           /                  / 2    2\ |                                                                                                             |         /                  / 2    2\           /                  / 2    2\ |                                                        \/         polar_lift\r  - y /  \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                       \/         polar_lift\r  - y /  \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                                                                                                                                                                                                                                                                                                                                                
 |  \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                                                                                                                                                                                                                                                    \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                               \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                                                      \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                                                                             \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
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0

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

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 |  |                                                                                                                                                                                                                                                             3   /           / 2    2\  |                         4*x*y                                                  4*x*r                        |                                                                                                                                                                                                                                 2     /                x           / 2    2\ |                         4*x*y                                                  4*x*r                        |       2     /                x           / 2    2\ |                         4*x*y                                                  4*x*r                        |          2   /           / 2    2\  |                         4*x*y                                                  4*x*r                        |          2   /           / 2    2\  |                         4*x*y                                                  4*x*r                        |                                                                                                                                                                                                                                                                                          2     /                x           / 2    2\ |                         4*x*y                                                  4*x*r                        |      /           x            \        2     /                x           / 2    2\ |                         4*x*y                                                  4*x*r                        |      /           x            \                                                                                                                                                                                                                                                                                                        
 |  |                                                                                                                                                                                                                                                        2*I*x *\/  polar_lift\r  - y / *|- ---------------------------------------------------- + ----------------------------------------------------|                                                                                                                                                                                                                             pi*y *   /   -1 + ------------------- *\r  - y /*|- ---------------------------------------------------- + ----------------------------------------------------|   pi*r *   /   -1 + ------------------- *\r  - y /*|- ---------------------------------------------------- + ----------------------------------------------------|   2*I*x*y *\/  polar_lift\r  - y / *|- ---------------------------------------------------- + ----------------------------------------------------|   2*I*x*r *\/  polar_lift\r  - y / *|- ---------------------------------------------------- + ----------------------------------------------------|                                                                                                                                                                                                                                                                                     2*I*y *   /   -1 + ------------------- *\r  - y /*|- ---------------------------------------------------- + ----------------------------------------------------|*acosh|------------------------|   2*I*r *   /   -1 + ------------------- *\r  - y /*|- ---------------------------------------------------- + ----------------------------------------------------|*acosh|------------------------|                                                                                                                                                                                                                                                                                                        
 |  |                                                                                                                                                                                                                                                                                        |        __________________________                             __________________________                    |                                                                                                                                                                                                                                     /                   / 2    2\            |        __________________________                             __________________________                    |           /                   / 2    2\            |        __________________________                             __________________________                    |                                     |        __________________________                             __________________________                    |                                     |        __________________________                             __________________________                    |                                                                                                                                                                                                                                                                                              /                   / 2    2\            |        __________________________                             __________________________                    |      |   _____________________|            /                   / 2    2\            |        __________________________                             __________________________                    |      |   _____________________|                                                                                                                                                                                                                                                                                                        
 |  |                                                                                                                                                                                                                                                                                        |       /                2                                     /                2                             |                                                                                                                                                                                                                                   \/          polar_lift\r  - y /            |       /                2                                     /                2                             |         \/          polar_lift\r  - y /            |       /                2                                     /                2                             |                                     |       /                2                                     /                2                             |                                     |       /                2                                     /                2                             |                                                                                                                                                                                                                                                                                            \/          polar_lift\r  - y /            |       /                2                                     /                2                             |      |  /           / 2    2\ |          \/          polar_lift\r  - y /            |       /                2                                     /                2                             |      |  /           / 2    2\ |                                                    2 / 2    2\      /           x            \                                                                                              2 / 2    2\      /           x            \                                                                
 |  |                                                                                                                                                                                                                                                                                        |      /                x                     / 2    2\       /                x                     / 2    2\|                                                                                                                                                                                                                                                                              |      /                x                     / 2    2\       /                x                     / 2    2\|                                                    |      /                x                     / 2    2\       /                x                     / 2    2\|                                     |      /                x                     / 2    2\       /                x                     / 2    2\|                                     |      /                x                     / 2    2\       /                x                     / 2    2\|                                                                                                                                                                                                                                                                                                                                       |      /                x                     / 2    2\       /                x                     / 2    2\|      \\/  polar_lift\r  - y / /                                                     |      /                x                     / 2    2\       /                x                     / 2    2\|      \\/  polar_lift\r  - y / /                                             2*I*x*y *\r  - y /*acosh|------------------------|                                                                                       2*I*x*r *\r  - y /*acosh|------------------------|                                                                
 |  |                                    _____________________                                                             _____________________                                                             _____________________                                                           |     /   -1 + ------------------- *polar_lift\r  - y /      /   -1 + ------------------- *polar_lift\r  - y /|                                                                                                                                                                                                                                                                              |     /   -1 + ------------------- *polar_lift\r  - y /      /   -1 + ------------------- *polar_lift\r  - y /|                                                    |     /   -1 + ------------------- *polar_lift\r  - y /      /   -1 + ------------------- *polar_lift\r  - y /|                                     |     /   -1 + ------------------- *polar_lift\r  - y /      /   -1 + ------------------- *polar_lift\r  - y /|                                     |     /   -1 + ------------------- *polar_lift\r  - y /      /   -1 + ------------------- *polar_lift\r  - y /|                                                                                                                                                                                                                                                                                                                                       |     /   -1 + ------------------- *polar_lift\r  - y /      /   -1 + ------------------- *polar_lift\r  - y /|                                                                                     |     /   -1 + ------------------- *polar_lift\r  - y /      /   -1 + ------------------- *polar_lift\r  - y /|                                                                                                     |   _____________________|                                                                                                               |   _____________________|                                                                
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 |  |                          6*I*x *\/  polar_lift\r  - y /                                                    2*I*y *\/  polar_lift\r  - y /                                                    2*I*r *\/  polar_lift\r  - y /                                                            \  \/          polar_lift\r  - y /                        \/          polar_lift\r  - y /                     /                                                2*I*y *\r  - y /                                                                                             2*I*r *\r  - y /                                                                                                 \  \/          polar_lift\r  - y /                        \/          polar_lift\r  - y /                     /                                                    \  \/          polar_lift\r  - y /                        \/          polar_lift\r  - y /                     /                                     \  \/          polar_lift\r  - y /                        \/          polar_lift\r  - y /                     /                                     \  \/          polar_lift\r  - y /                        \/          polar_lift\r  - y /                     /                                                             pi*x*y *\r  - y /                                                                                                                        pi*x*r *\r  - y /                                                                                                                \  \/          polar_lift\r  - y /                        \/          polar_lift\r  - y /                     /                                                                                     \  \/          polar_lift\r  - y /                        \/          polar_lift\r  - y /                     /                                                                                                     \\/  polar_lift\r  - y / /                                                                                                               \\/  polar_lift\r  - y / /                                                    x           
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 |  |               __________________________              __________________________                __________________________              __________________________                __________________________              __________________________                                                                                                                   2                                 /             __________________________              __________________________\                            /             __________________________              __________________________\                                                                                                                                                    2                                                                                                                                                                  2                                                                                                                                                           2                                                                                                                                                   2                                        __________________________ /             __________________________              __________________________\                             __________________________ /             __________________________              __________________________\                                                                                                                                                                2                                                                                                                                                                                                   2                                                                __________________________ /             __________________________              __________________________\                             __________________________ /             __________________________              __________________________\                          | 2    2|       
 |  |              /                2                      /                2                        /                2                      /                2                        /                2                      /                2                                           /             __________________________              __________________________\                                  |            /                2                      /                2         |    _____________________   |            /                2                      /                2         |    _____________________                                          /             __________________________              __________________________\                                                                                  /             __________________________              __________________________\                                                                           /             __________________________              __________________________\                                                                   /             __________________________              __________________________\                                        /                2          |            /                2                      /                2         |                            /                2          |            /                2                      /                2         |                                                                               /             __________________________              __________________________\                                                                                                                   /             __________________________              __________________________\                                                                /                2          |            /                2                      /                2         |                            /                2          |            /                2                      /                2         |                          |r  - y |       
 |  |       2     /                x                2     /                x                  2     /                x                2     /                x                  2     /                x                2     /                x                                            |            /                2                      /                2         |                                  |     2     /                x                2     /                x          |   /           / 2    2\    |     2     /                x                2     /                x          |   /           / 2    2\                                           |            /                2                      /                2         |                                                                                  |            /                2                      /                2         |                                                                           |            /                2                      /                2         |                                                                   |            /                2                      /                2         |                                       /                x           |     2     /                x                2     /                x          |           / 2    2\       /                x           |     2     /                x                2     /                x          |           / 2    2\                                                           |            /                2                      /                2         |                                                                                                                   |            /                2                      /                2         |                                                               /                x           |     2     /                x                2     /                x          |           / 2    2\       /                x           |     2     /                x                2     /                x          |           / 2    2\                      
 |  |  - 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + -------------------    - 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + -------------------    - 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + -------------------                                   |     2     /                x                2     /                x          |                                  |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |*\/  polar_lift\r  - y /    |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |*\/  polar_lift\r  - y /                                           |     2     /                x                2     /                x          |                                                                                  |     2     /                x                2     /                x          |                                                                           |     2     /                x                2     /                x          |                                                                   |     2     /                x                2     /                x          |                                      /   -1 + ------------------- *|- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |*polar_lift\r  - y /      /   -1 + ------------------- *|- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |*polar_lift\r  - y /                                                           |     2     /                x                2     /                x          |                                                                                                                   |     2     /                x                2     /                x          |                                                              /   -1 + ------------------- *|- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |*polar_lift\r  - y /      /   -1 + ------------------- *|- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |*polar_lift\r  - y /                      
 |  |           /                   / 2    2\           /                   / 2    2\             /                   / 2    2\           /                   / 2    2\             /                   / 2    2\           /                   / 2    2\                                   |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                  |         /                   / 2    2\           /                   / 2    2\ |                            |         /                   / 2    2\           /                   / 2    2\ |                                                                   |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                                                                  |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                                                           |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                                                   |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                     /                   / 2    2\  |         /                   / 2    2\           /                   / 2    2\ |                         /                   / 2    2\  |         /                   / 2    2\           /                   / 2    2\ |                                                                               |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                                                                                                   |- 4*r *   /   -1 + -------------------  + 4*y *   /   -1 + ------------------- |                                                             /                   / 2    2\  |         /                   / 2    2\           /                   / 2    2\ |                         /                   / 2    2\  |         /                   / 2    2\           /                   / 2    2\ |                                          
 |  |         \/          polar_lift\r  - y /         \/          polar_lift\r  - y /           \/          polar_lift\r  - y /         \/          polar_lift\r  - y /           \/          polar_lift\r  - y /         \/          polar_lift\r  - y /                                   |         /                   / 2    2\           /                   / 2    2\ |                                  \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                            \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                   |         /                   / 2    2\           /                   / 2    2\ |                                                                                  |         /                   / 2    2\           /                   / 2    2\ |                                                                           |         /                   / 2    2\           /                   / 2    2\ |                                                                   |         /                   / 2    2\           /                   / 2    2\ |                                   \/          polar_lift\r  - y /  \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                       \/          polar_lift\r  - y /  \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                               |         /                   / 2    2\           /                   / 2    2\ |                                                                                                                   |         /                   / 2    2\           /                   / 2    2\ |                                                           \/          polar_lift\r  - y /  \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                       \/          polar_lift\r  - y /  \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                          
 |  |                                                                                                                                                                                                                                                                                       \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                                                                                                                                                                                                                                                   \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                                  \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                           \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                   \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                                                                                                                                                                                                                                                                                                                             \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                                                                   \       \/          polar_lift\r  - y /         \/          polar_lift\r  - y / /                                                                                                                                                                                                                                                                                                                                                                
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 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             _________________________                                                                                                                                                                   _________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            _____________________ /                              2                                                     2                      \                                                                                                                                                                                                                                 _____________________ /                              2                                                     2                      \           _____________________ /                              2                                                     2                      \           /               2                    /                              2                                                     2                      \                                          /               2                    /                              2                                                     2                      \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       3   /           / 2    2\  |                         2*x*r                                                 2*x*y                       |                                                                                                                                                                                                                            2   /           / 2    2\  |                         2*x*r                                                 2*x*y                       |      2   /           / 2    2\  |                         2*x*r                                                 2*x*y                       |    2     /               x           / 2    2\ |                         2*x*r                                                 2*x*y                       |     /           x            \    2     /               x           / 2    2\ |                         2*x*r                                                 2*x*y                       |     /           x            \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      x *\/  polar_lift\r  - y / *|- --------------------------------------------------- + ---------------------------------------------------|                                                                                                                                                                                                                         x*y *\/  polar_lift\r  - y / *|- --------------------------------------------------- + ---------------------------------------------------|   x*r *\/  polar_lift\r  - y / *|- --------------------------------------------------- + ---------------------------------------------------|   y *   /   1 - ------------------- *\r  - y /*|- --------------------------------------------------- + ---------------------------------------------------|*asin|------------------------|   r *   /   1 - ------------------- *\r  - y /*|- --------------------------------------------------- + ---------------------------------------------------|*asin|------------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |        _________________________                             _________________________                    |                                                                                                                                                                                                                                                       |        _________________________                             _________________________                    |                                 |        _________________________                             _________________________                    |        /                  / 2    2\            |        _________________________                             _________________________                    |     |   _____________________|        /                  / 2    2\            |        _________________________                             _________________________                    |     |   _____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |       /               2                                     /               2                             |                                                                                                                                                                                                                                                       |       /               2                                     /               2                             |                                 |       /               2                                     /               2                             |      \/         polar_lift\r  - y /            |       /               2                                     /               2                             |     |  /           / 2    2\ |      \/         polar_lift\r  - y /            |       /               2                                     /               2                             |     |  /           / 2    2\ |                                                 2 / 2    2\     /           x            \                                                                                            2 / 2    2\     /           x            \                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |      /               x                     / 2    2\       /               x                     / 2    2\|                                                                                                                                                                                                                                                       |      /               x                     / 2    2\       /               x                     / 2    2\|                                 |      /               x                     / 2    2\       /               x                     / 2    2\|                                                |      /               x                     / 2    2\       /               x                     / 2    2\|     \\/  polar_lift\r  - y / /                                                |      /               x                     / 2    2\       /               x                     / 2    2\|     \\/  polar_lift\r  - y / /                                              x*r *\r  - y /*asin|------------------------|                                                                                         x*y *\r  - y /*asin|------------------------|                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                                     _____________________                                                           _____________________                                                            _____________________                                                       |     /   1 - ------------------- *polar_lift\r  - y /      /   1 - ------------------- *polar_lift\r  - y /|                                                                                                                                                                                                                                                       |     /   1 - ------------------- *polar_lift\r  - y /      /   1 - ------------------- *polar_lift\r  - y /|                                 |     /   1 - ------------------- *polar_lift\r  - y /      /   1 - ------------------- *polar_lift\r  - y /|                                                |     /   1 - ------------------- *polar_lift\r  - y /      /   1 - ------------------- *polar_lift\r  - y /|                                                                               |     /   1 - ------------------- *polar_lift\r  - y /      /   1 - ------------------- *polar_lift\r  - y /|                                                                                                |   _____________________|                                                                                                            |   _____________________|                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                                2   /           / 2    2\                                                       2   /           / 2    2\                                                        2   /           / 2    2\                                                        |    /                  / 2    2\                          /                  / 2    2\                     |                                                  2 / 2    2\                                                                                                2 / 2    2\                                                                               |    /                  / 2    2\                          /                  / 2    2\                     |                                 |    /                  / 2    2\                          /                  / 2    2\                     |                                                |    /                  / 2    2\                          /                  / 2    2\                     |                                                                               |    /                  / 2    2\                          /                  / 2    2\                     |                                                                                                |  /           / 2    2\ |                                                                                                            |  /           / 2    2\ |                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                               y *\/  polar_lift\r  - y /                                                      r *\/  polar_lift\r  - y /                                                     3*x *\/  polar_lift\r  - y /                                                        \  \/         polar_lift\r  - y /                        \/         polar_lift\r  - y /                     /                                                 y *\r  - y /                                                                                               r *\r  - y /                                                                               \  \/         polar_lift\r  - y /                        \/         polar_lift\r  - y /                     /                                 \  \/         polar_lift\r  - y /                        \/         polar_lift\r  - y /                     /                                                \  \/         polar_lift\r  - y /                        \/         polar_lift\r  - y /                     /                                                                               \  \/         polar_lift\r  - y /                        \/         polar_lift\r  - y /                     /                                                                                                \\/  polar_lift\r  - y / /                                                                                                            \\/  polar_lift\r  - y / /                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                      ----------------------------------------------------------------------------- - ----------------------------------------------------------------------------- + ----------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------------------------------------------                                                                                                                                                                                                                                                                                                                                            otherwise       
 |  |                                                                                                                                                                                                                                                                                                                                                   _________________________              _________________________                _________________________              _________________________                _________________________              _________________________                                                                                                               2                               /             _________________________              _________________________\                            /             _________________________              _________________________\                                                                                                                                         2                                                                                                                                             2                                                                                                                                                                    2                                                                                                                                                                                           2                                                             _________________________ /             _________________________              _________________________\                             _________________________ /             _________________________              _________________________\                                                                                                                                                                                                                                                                                                                                                                                
 |  |                                                                                                                                                                                                                                                                                                                                                  /               2                      /               2                        /               2                      /               2                        /               2                      /               2                                         /             _________________________              _________________________\                                |            /               2                      /               2         |    _____________________   |            /               2                      /               2         |    _____________________                                 /             _________________________              _________________________\                                                               /             _________________________              _________________________\                                                                                      /             _________________________              _________________________\                                                                                                             /             _________________________              _________________________\                                                             /               2          |            /               2                      /               2         |                            /               2          |            /               2                      /               2         |                                                                                                                                                                                                                                                                                                                                                                                
 |  |                                                                                                                                                                                                                                                                                                                                           2     /               x                2     /               x                  2     /               x                2     /               x                  2     /               x                2     /               x                                          |            /               2                      /               2         |                                |     2     /               x                2     /               x          |   /           / 2    2\    |     2     /               x                2     /               x          |   /           / 2    2\                                  |            /               2                      /               2         |                                                               |            /               2                      /               2         |                                                                                      |            /               2                      /               2         |                                                                                                             |            /               2                      /               2         |                                                            /               x           |     2     /               x                2     /               x          |           / 2    2\       /               x           |     2     /               x                2     /               x          |           / 2    2\                                                                                                                                                                                                                                                                                                                                                            
 |  |                                                                                                                                                                                                                                                                                                                                      - 2*r *   /   1 - -------------------  + 2*y *   /   1 - -------------------    - 2*r *   /   1 - -------------------  + 2*y *   /   1 - -------------------    - 2*r *   /   1 - -------------------  + 2*y *   /   1 - -------------------                                 |     2     /               x                2     /               x          |                                |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |*\/  polar_lift\r  - y /    |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |*\/  polar_lift\r  - y /                                  |     2     /               x                2     /               x          |                                                               |     2     /               x                2     /               x          |                                                                                      |     2     /               x                2     /               x          |                                                                                                             |     2     /               x                2     /               x          |                                                           /   1 - ------------------- *|- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |*polar_lift\r  - y /      /   1 - ------------------- *|- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |*polar_lift\r  - y /                                                                                                                                                                                                                                                                                                                                                            
 |  |                                                                                                                                                                                                                                                                                                                                               /                  / 2    2\           /                  / 2    2\             /                  / 2    2\           /                  / 2    2\             /                  / 2    2\           /                  / 2    2\                                 |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |                                |         /                  / 2    2\           /                  / 2    2\ |                            |         /                  / 2    2\           /                  / 2    2\ |                                                          |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |                                                               |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |                                                                                      |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |                                                                                                             |- 2*r *   /   1 - -------------------  + 2*y *   /   1 - ------------------- |                                                          /                  / 2    2\  |         /                  / 2    2\           /                  / 2    2\ |                         /                  / 2    2\  |         /                  / 2    2\           /                  / 2    2\ |                                                                                                                                                                                                                                                                                                                                                                                
 |  |                                                                                                                                                                                                                                                                                                                                             \/         polar_lift\r  - y /         \/         polar_lift\r  - y /           \/         polar_lift\r  - y /         \/         polar_lift\r  - y /           \/         polar_lift\r  - y /         \/         polar_lift\r  - y /                                 |         /                  / 2    2\           /                  / 2    2\ |                                \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                            \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                          |         /                  / 2    2\           /                  / 2    2\ |                                                               |         /                  / 2    2\           /                  / 2    2\ |                                                                                      |         /                  / 2    2\           /                  / 2    2\ |                                                                                                             |         /                  / 2    2\           /                  / 2    2\ |                                                        \/         polar_lift\r  - y /  \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                       \/         polar_lift\r  - y /  \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                                                                                                                                                                                                                                                                                                                                                
 |  \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                                                                                                                                                                                                                                                    \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                               \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                                                      \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                                                                             \       \/         polar_lift\r  - y /         \/         polar_lift\r  - y / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
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0

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

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

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