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Integral of d{x}:
Integral of 6x^5
Integral of x(x^2+3)
Integral of 1/(x(1+x^2))
Integral of e^(-x)*dx
Identical expressions
√(one -x²-y²)
√(1 minus x² minus y²)
√(one minus x² minus y²)
√1-x²-y²
√(1-x²-y²)dx
Similar expressions
√(1-x²+y²)
√(1+x²-y²)

Integral of √(1-x²-y²) dy

The solution

You have entered [src]

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0

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

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

The answer (Indefinite) [src]

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                             ||                    \/          polar_lift\1 - x /                                                   2*I*y *\/  polar_lift\1 - x /                                                2*I*y*\/  polar_lift\1 - x /                                               \/          polar_lift\1 - x /                                                2*I*y*x *\/  polar_lift\1 - x /                                                                           \\/  polar_lift\1 - x / /                                                        \\/  polar_lift\1 - x / /      |   y   |    |
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 |   /      2    2           ||      \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /         \/          polar_lift\1 - x /         \/          polar_lift\1 - x /          \/          polar_lift\1 - x /         \/          polar_lift\1 - x /                      |
 | \/  1 - x  - y   dy = C + |<                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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                             ||                                                                                                            y *\/  polar_lift\1 - x /                                                  y*\/  polar_lift\1 - x /                                                  y*x *\/  polar_lift\1 - x /                                                                       \\/  polar_lift\1 - x / /                                                    \\/  polar_lift\1 - x / /                                                                                                         |
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                             \\                                                                                         \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /                                                                                                          /

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

Simplify

The answer [src]

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

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

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 |  |                                                                                                                                                                                                                                                                                                                                                     /               y           /     2\ |                       4*y                                                 4*y*x                       |                                                                                                               3   /           /     2\  |                       4*y                                                 4*y*x                       |           /           /     2\  |                       4*y                                                 4*y*x                       |       2     /               y           /     2\ |                       4*y                                                 4*y*x                       |                                                                                                                                            2   /           /     2\  |                       4*y                                                 4*y*x                       |           /               y           /     2\ |                       4*y                                                 4*y*x                       |      /           y           \                                                                                                                                          2     /               y           /     2\ |                       4*y                                                 4*y*x                       |      /           y           \                                                                                                                                                                                                                                                                                          
 |  |                                                                                                                                                                                                                                                                                                                                              pi*   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|                                                                                                          2*I*y *\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|   2*I*y*\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|   pi*x *   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|                                                                                                                                     2*I*y*x *\/  polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------|   2*I*   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                     2*I*x *   /   -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                                                                                                                                                                          
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 |  |                                                                                                                                                                                                                                                                                                                                                                                          |    /               y                     /     2\       /               y                     /     2\|                                                                                                                                         |    /               y                     /     2\       /               y                     /     2\|                                 |    /               y                     /     2\       /               y                     /     2\|                                                  |    /               y                     /     2\       /               y                     /     2\|                                                                                                                                                                      |    /               y                     /     2\       /               y                     /     2\|                                                |    /               y                     /     2\       /               y                     /     2\|      \\/  polar_lift\1 - x / /                                                                                                                                                                                     |    /               y                     /     2\       /               y                     /     2\|      \\/  polar_lift\1 - x / /                                            2*I*y*\1 - x /*acosh|-----------------------|                                                                                    2*I*y*x *\1 - x /*acosh|-----------------------|                                                             
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 |  |                       2*I*\/  polar_lift\1 - x /                                                  6*I*y *\/  polar_lift\1 - x /                                                2*I*x *\/  polar_lift\1 - x /                                                                      2*I*\1 - x /                                                                                           \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                             2*I*x *\1 - x /                                                                             \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                 \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                  \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                            pi*y*\1 - x /                                                                                             \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                                                          pi*y*x *\1 - x /                                                                                                          \\/          polar_lift\1 - x /                       \/          polar_lift\1 - x /                    /                                                                                               \\/  polar_lift\1 - x / /                                                                                                           \\/  polar_lift\1 - x / /                                                 y           
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 |  |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------    |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*\/  polar_lift\1 - x /                                        |        /               y                2     /               y          |                                        |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*\/  polar_lift\1 - x /                                 |        /               y                2     /               y          |                                                               |        /               y                2     /               y          |                                                                      |        /               y                2     /               y          |                                             /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /                                 |        /               y                2     /               y          |                                                                                       |        /               y                2     /               y          |                                                           /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /                                                        |        /               y                2     /               y          |                                                             /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /      /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |*polar_lift\1 - x /                      
 |  |      /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\    |      /                   /     2\           /                   /     2\ |                                                               |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                        |      /                   /     2\           /                   /     2\ |                                                        |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                               |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                      |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                            /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                    |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                                                       |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                          /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                                           |- 4*   /   -1 + ------------------  + 4*x *   /   -1 + ------------------ |                                                            /                   /     2\  |      /                   /     2\           /                   /     2\ |                        /                   /     2\  |      /                   /     2\           /                   /     2\ |                                         
 |  |    \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /        \/          polar_lift\1 - x /         \/          polar_lift\1 - x /    \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                               |      /                   /     2\           /                   /     2\ |                                        \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                        |      /                   /     2\           /                   /     2\ |                                                               |      /                   /     2\           /                   /     2\ |                                                                      |      /                   /     2\           /                   /     2\ |                                          \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                    |      /                   /     2\           /                   /     2\ |                                                                                       |      /                   /     2\           /                   /     2\ |                                                        \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                           |      /                   /     2\           /                   /     2\ |                                                          \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                      \/          polar_lift\1 - x /  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                         
 |  |                                                                                                                                                                                                                                                                                                                                                                                  \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                                                                                            \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                               \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                      \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                                                                                                                          \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                       \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                                                                                                                                                               \    \/          polar_lift\1 - x /         \/          polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                 
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 |  |                                                                                                                                                                                                                                                                                                                                        \/  polar_lift\1 - x /                                                    x *\/  polar_lift\1 - x /                                                                         1 - x                                                                        3*y *\/  polar_lift\1 - x /                                                                     x *\1 - x /                                                                          \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                             \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                           \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                                                          \\/  polar_lift\1 - x / /                                                                                        \  \/         polar_lift\1 - x /                       \/         polar_lift\1 - x /                    /                                                                                           \\/  polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                              
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 |  |                                                                                                                                                                                                                                                                                                                - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------    - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------    |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*\/  polar_lift\1 - x /    - 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------    |- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*\/  polar_lift\1 - x /                                |        /              y                2     /              y          |                                                             |        /              y                2     /              y          |                                                             |        /              y                2     /              y          |                                                                                    |        /              y                2     /              y          |                                                         /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*polar_lift\1 - x /                                                      |        /              y                2     /              y          |                                                           /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*x *   /   1 - ------------------ |*polar_lift\1 - x /                                                                                                                                                                                                                                                                                                                                     
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 |  |                                                                                                                                                                                                                                                                                                                    \/         polar_lift\1 - x /         \/         polar_lift\1 - x /        \/         polar_lift\1 - x /         \/         polar_lift\1 - x /    \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                               \/         polar_lift\1 - x /         \/         polar_lift\1 - x /    \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                       |      /                  /     2\           /                  /     2\ |                                                             |      /                  /     2\           /                  /     2\ |                                                             |      /                  /     2\           /                  /     2\ |                                                                                    |      /                  /     2\           /                  /     2\ |                                                      \/         polar_lift\1 - x /  \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                         |      /                  /     2\           /                  /     2\ |                                                        \/         polar_lift\1 - x /  \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                        
 |  \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                             \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                             \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                    \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                                        \    \/         polar_lift\1 - x /         \/         polar_lift\1 - x / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         
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/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
0

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

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

Use the examples entering the upper and lower limits of integration.

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Integral of √(1-x²-y²) dy

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Piecewise:

The solution