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Integral of √(1-x²-y²) dy
The solution
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[src]
1 / | | _____________ | / 2 2 | \/ 1 - x - y dy | / 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]
// _________________________ _________________________ \
|| _________________________ _________________________ / 2 / 2 |
|| / 2 / 2 / y / 2\ / y \ 2 / y / 2\ / y \ |
|| / y / 2\ 2 / y / 2\ 2*I* / -1 + ------------------ *\1 - x /*acosh|-----------------------| 2*I*x * / -1 + ------------------ *\1 - x /*acosh|-----------------------| |
|| pi* / -1 + ------------------ *\1 - x / ____________________ ____________________ pi*x * / -1 + ------------------ *\1 - x / ____________________ / / 2\ | ____________________| / / 2\ | ____________________| |
|| / / 2\ 3 / / 2\ / / 2\ / / 2\ 2 / / 2\ \/ polar_lift\1 - x / | / / 2\ | \/ polar_lift\1 - x / | / / 2\ | | 2 | |
|| \/ 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 | |
||- -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + -------------------------------------------------------------------------- + -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + --------------------------------------------------------------------------- - ------------------------------------------------------------------------------ for |-------| > 1|
|| _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ | 2| |
|| / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 |-1 + x | |
/ || / y 2 / y / y 2 / y / y 2 / y / y 2 / y / y 2 / y / y 2 / y / y 2 / y |
| || - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |
| _____________ || / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ |
| / 2 2 || \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / |
| \/ 1 - x - y dy = C + |< |
| || ________________________ ________________________ |
/ || / 2 / 2 |
|| / y / 2\ / y \ 2 / y / 2\ / y \ |
|| / 1 - ------------------ *\1 - x /*asin|-----------------------| x * / 1 - ------------------ *\1 - x /*asin|-----------------------| |
|| ____________________ ____________________ ____________________ / / 2\ | ____________________| / / 2\ | ____________________| |
|| 3 / / 2\ / / 2\ 2 / / 2\ \/ polar_lift\1 - x / | / / 2\ | \/ polar_lift\1 - x / | / / 2\ | |
|| y *\/ polar_lift\1 - x / y*\/ polar_lift\1 - x / y*x *\/ polar_lift\1 - x / \\/ polar_lift\1 - x / / \\/ polar_lift\1 - x / / |
|| ------------------------------------------------------------------------ - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ otherwise |
|| ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ |
|| / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 |
|| / y 2 / y / y 2 / y / y 2 / y / y 2 / y / y 2 / y |
|| - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |
|| / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ |
\\ \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / /
$$\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}$$
The answer
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1 / | | / _________________________ _________________________ _________________________ _________________________ | | / 2 / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | / 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|-----------------------| | | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | ____________________| / / 2\ | _________________________ _________________________ | | ____________________| | | \/ 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 \ | | | / 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|-----------------------| | | ____________________ ____________________ ____________________ | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | ____________________| | ____________________| | | / / 2\ 2 / / 2\ 2 / / 2\ / 2\ | / / 2\ / / 2\ | 2 / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | 2 / 2\ | / / 2\ / / 2\ | | / / 2\ | | / / 2\ | 2 | | 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 | |-------------------------------------------------------------------------- - -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- for --------- > 1 | | _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ / _________________________ _________________________\ 2 / _________________________ _________________________\ 2 2 2 _________________________ / _________________________ _________________________\ 2 2 _________________________ / _________________________ _________________________\ 2 _________________________ / _________________________ _________________________\ _________________________ / _________________________ _________________________\ | 2| | | / 2 / 2 / 2 / 2 / 2 / 2 | / 2 / 2 | ____________________ / _________________________ _________________________\ | / 2 / 2 | ____________________ / _________________________ _________________________\ / _________________________ _________________________\ / _________________________ _________________________\ / 2 | / 2 / 2 | / _________________________ _________________________\ / _________________________ _________________________\ / 2 | / 2 / 2 | / _________________________ _________________________\ / 2 | / 2 / 2 | / 2 | / 2 / 2 | |-1 + x | | | / y 2 / y / y 2 / y / y 2 / y | / y 2 / y | / / 2\ | / 2 / 2 | | / y 2 / y | / / 2\ | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | / y | / y 2 / y | / 2\ | / 2 / 2 | | / 2 / 2 | / y | / y 2 / y | / 2\ | / 2 / 2 | / y | / y 2 / y | / 2\ / y | / y 2 / y | / 2\ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*\/ polar_lift\1 - x / | / 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 | | ________________________ ________________________ | | ____________________ / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | 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\ | ________________________ ________________________ | | ____________________| | | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - x / | / 2 / 2 | | / / 2\ | / 2\ / y \ \/ polar_lift\1 - x / | / 2 / 2 | | / / 2\ | 2 / 2\ / y \ | | | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / 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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1 / | | / _________________________ _________________________ _________________________ _________________________ | | / 2 / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | / 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|-----------------------| | | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | ____________________| / / 2\ | _________________________ _________________________ | | ____________________| | | \/ 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 \ | | | / 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|-----------------------| | | ____________________ ____________________ ____________________ | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | ____________________| | ____________________| | | / / 2\ 2 / / 2\ 2 / / 2\ / 2\ | / / 2\ / / 2\ | 2 / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | 2 / 2\ | / / 2\ / / 2\ | | / / 2\ | | / / 2\ | 2 | | 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 | |-------------------------------------------------------------------------- - -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- for --------- > 1 | | _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ / _________________________ _________________________\ 2 / _________________________ _________________________\ 2 2 2 _________________________ / _________________________ _________________________\ 2 2 _________________________ / _________________________ _________________________\ 2 _________________________ / _________________________ _________________________\ _________________________ / _________________________ _________________________\ | 2| | | / 2 / 2 / 2 / 2 / 2 / 2 | / 2 / 2 | ____________________ / _________________________ _________________________\ | / 2 / 2 | ____________________ / _________________________ _________________________\ / _________________________ _________________________\ / _________________________ _________________________\ / 2 | / 2 / 2 | / _________________________ _________________________\ / _________________________ _________________________\ / 2 | / 2 / 2 | / _________________________ _________________________\ / 2 | / 2 / 2 | / 2 | / 2 / 2 | |-1 + x | | | / y 2 / y / y 2 / y / y 2 / y | / y 2 / y | / / 2\ | / 2 / 2 | | / y 2 / y | / / 2\ | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | / y | / y 2 / y | / 2\ | / 2 / 2 | | / 2 / 2 | / y | / y 2 / y | / 2\ | / 2 / 2 | / y | / y 2 / y | / 2\ / y | / y 2 / y | / 2\ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*\/ polar_lift\1 - x / | / 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 | | ________________________ ________________________ | | ____________________ / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | 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\ | ________________________ ________________________ | | ____________________| | | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - x / | / 2 / 2 | | / / 2\ | / 2\ / y \ \/ polar_lift\1 - x / | / 2 / 2 | | / / 2\ | 2 / 2\ / y \ | | | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / 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$$
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.