Sr Examen
Integral de sqrt(1-x^2-y^2) dx
Solución
Ha introducido
[src]
1 / | | _____________ | / 2 2 | \/ 1 - x - y dx | / 0
$$\int\limits_{0}^{1} \sqrt{- y^{2} + \left(1 - x^{2}\right)}\, dx$$
Integral(sqrt(1 - x^2 - y^2), (x, 0, 1))
Respuesta (Indefinida)
[src]
// _________________________ _________________________ \
|| _________________________ _________________________ / 2 / 2 |
|| / 2 / 2 / x / 2\ / x \ 2 / x / 2\ / x \ |
|| / x / 2\ 2 / x / 2\ 2*I* / -1 + ------------------ *\1 - y /*acosh|-----------------------| 2*I*y * / -1 + ------------------ *\1 - y /*acosh|-----------------------| |
|| pi* / -1 + ------------------ *\1 - y / ____________________ ____________________ pi*y * / -1 + ------------------ *\1 - y / ____________________ / / 2\ | ____________________| / / 2\ | ____________________| |
|| / / 2\ 3 / / 2\ / / 2\ / / 2\ 2 / / 2\ \/ polar_lift\1 - y / | / / 2\ | \/ polar_lift\1 - y / | / / 2\ | | 2 | |
|| \/ polar_lift\1 - y / 2*I*x *\/ polar_lift\1 - y / 2*I*x*\/ polar_lift\1 - y / \/ polar_lift\1 - y / 2*I*x*y *\/ polar_lift\1 - y / \\/ polar_lift\1 - y / / \\/ polar_lift\1 - y / / | x | |
||- -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + -------------------------------------------------------------------------- + -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + --------------------------------------------------------------------------- - ------------------------------------------------------------------------------ for |-------| > 1|
|| _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ | 2| |
|| / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 |-1 + y | |
/ || / x 2 / x / x 2 / x / x 2 / x / x 2 / x / x 2 / x / x 2 / x / x 2 / x |
| || - 4* / -1 + ------------------ + 4*y * / -1 + ------------------ - 4* / -1 + ------------------ + 4*y * / -1 + ------------------ - 4* / -1 + ------------------ + 4*y * / -1 + ------------------ - 4* / -1 + ------------------ + 4*y * / -1 + ------------------ - 4* / -1 + ------------------ + 4*y * / -1 + ------------------ - 4* / -1 + ------------------ + 4*y * / -1 + ------------------ - 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |
| _____________ || / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ |
| / 2 2 || \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / |
| \/ 1 - x - y dx = C + |< |
| || ________________________ ________________________ |
/ || / 2 / 2 |
|| / x / 2\ / x \ 2 / x / 2\ / x \ |
|| / 1 - ------------------ *\1 - y /*asin|-----------------------| y * / 1 - ------------------ *\1 - y /*asin|-----------------------| |
|| ____________________ ____________________ ____________________ / / 2\ | ____________________| / / 2\ | ____________________| |
|| 3 / / 2\ / / 2\ 2 / / 2\ \/ polar_lift\1 - y / | / / 2\ | \/ polar_lift\1 - y / | / / 2\ | |
|| x *\/ polar_lift\1 - y / x*\/ polar_lift\1 - y / x*y *\/ polar_lift\1 - y / \\/ polar_lift\1 - y / / \\/ polar_lift\1 - y / / |
|| ------------------------------------------------------------------------ - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ otherwise |
|| ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ |
|| / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 |
|| / x 2 / x / x 2 / x / x 2 / x / x 2 / x / x 2 / x |
|| - 2* / 1 - ------------------ + 2*y * / 1 - ------------------ - 2* / 1 - ------------------ + 2*y * / 1 - ------------------ - 2* / 1 - ------------------ + 2*y * / 1 - ------------------ - 2* / 1 - ------------------ + 2*y * / 1 - ------------------ - 2* / 1 - ------------------ + 2*y * / 1 - ------------------ |
|| / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ |
\\ \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / /
$$\int \sqrt{- y^{2} + \left(1 - x^{2}\right)}\, dx = C + \begin{cases} - \frac{2 i x^{3} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} + \frac{2 i x \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} - \frac{2 i y^{2} \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} + \frac{\pi y^{2} \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} + \frac{2 i \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} - \frac{\pi \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2} - 1}}\right| > 1 \\\frac{x^{3} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} + \frac{x y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{x \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} + \frac{y^{2} \left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{\left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}$$
Respuesta
[src]
1 / | | / _________________________ _________________________ _________________________ _________________________ | | / 2 / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | / x / 2\ | 4*x 4*x*y | 3 / / 2\ | 4*x 4*x*y | / / 2\ | 4*x 4*x*y | 2 / x / 2\ | 4*x 4*x*y | 2 / / 2\ | 4*x 4*x*y | / x / 2\ | 4*x 4*x*y | / x \ 2 / x / 2\ | 4*x 4*x*y | / x \ | | pi* / -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------| 2*I*x *\/ polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------| 2*I*x*\/ polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------| pi*y * / -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------| 2*I*x*y *\/ polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------| 2*I* / -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------| 2*I*y * / -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------| | | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | ____________________| / / 2\ | _________________________ _________________________ | | ____________________| | | \/ polar_lift\1 - y / | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - y / | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - y / | / 2 / 2 | | / / 2\ | \/ polar_lift\1 - y / | / 2 / 2 | | / / 2\ | / 2\ / x \ 2 / 2\ / x \ | | | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| \\/ polar_lift\1 - y / / | / x / 2\ / x / 2\| \\/ polar_lift\1 - y / / 2*I*x*\1 - y /*acosh|-----------------------| 2*I*x*y *\1 - y /*acosh|-----------------------| | | ____________________ ____________________ ____________________ | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | ____________________| | ____________________| | | / / 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 - y / 6*I*x *\/ polar_lift\1 - y / 2*I*y *\/ polar_lift\1 - y / 2*I*\1 - y / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / 2*I*y *\1 - y / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / pi*x*\1 - y / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / pi*x*y *\1 - y / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / \\/ polar_lift\1 - y / / \\/ polar_lift\1 - y / / x | |-------------------------------------------------------------------------- - -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- 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 + y | | | / x 2 / x / x 2 / x / x 2 / x | / x 2 / x | / / 2\ | / 2 / 2 | | / 2 / 2 | | / x 2 / x | / / 2\ | / 2 / 2 | | / 2 / 2 | / x | / x 2 / x | / 2\ | / 2 / 2 | | / 2 / 2 | / x | / x 2 / x | / 2\ | / 2 / 2 | / x | / x 2 / x | / 2\ / x | / x 2 / x | / 2\ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ - 4* / -1 + ------------------ + 4*y * / -1 + ------------------ - 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |*\/ polar_lift\1 - y / | / x 2 / x | | / x 2 / x | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |*\/ polar_lift\1 - y / | / x 2 / x | | / x 2 / x | / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\1 - y / | / x 2 / x | | / x 2 / x | / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\1 - y / | / x 2 / x | / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\1 - y / / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\1 - y / | | / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ | | | \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / / 2\ / / 2\ | \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | | \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | < dx | | ________________________ ________________________ | | ____________________ / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | 3 / / 2\ | 2*x 2*x*y | / / 2\ | 2*x 2*x*y | 2 / / 2\ | 2*x 2*x*y | / x / 2\ | 2*x 2*x*y | / x \ 2 / x / 2\ | 2*x 2*x*y | / x \ | | x *\/ polar_lift\1 - y / *|- ------------------------------------------------- + -------------------------------------------------| x*\/ polar_lift\1 - y / *|- ------------------------------------------------- + -------------------------------------------------| x*y *\/ polar_lift\1 - y / *|- ------------------------------------------------- + -------------------------------------------------| / 1 - ------------------ *\1 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------| y * / 1 - ------------------ *\1 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------| | | | ________________________ ________________________ | | ________________________ ________________________ | | ________________________ ________________________ | / / 2\ | ________________________ ________________________ | | ____________________| / / 2\ | ________________________ ________________________ | | ____________________| | | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - y / | / 2 / 2 | | / / 2\ | / 2\ / x \ \/ polar_lift\1 - y / | / 2 / 2 | | / / 2\ | 2 / 2\ / x \ | | | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| \\/ polar_lift\1 - y / / x*\1 - y /*asin|-----------------------| | / x / 2\ / x / 2\| \\/ polar_lift\1 - y / / x*y *\1 - y /*asin|-----------------------| | | ____________________ ____________________ ____________________ | / 1 - ------------------ *polar_lift\1 - y / / 1 - ------------------ *polar_lift\1 - y /| | / 1 - ------------------ *polar_lift\1 - y / / 1 - ------------------ *polar_lift\1 - y /| | / 1 - ------------------ *polar_lift\1 - y / / 1 - ------------------ *polar_lift\1 - y /| | / 1 - ------------------ *polar_lift\1 - y / / 1 - ------------------ *polar_lift\1 - y /| | ____________________| | / 1 - ------------------ *polar_lift\1 - y / / 1 - ------------------ *polar_lift\1 - y /| | ____________________| | | / / 2\ 2 / / 2\ 2 2 / / 2\ | / / 2\ / / 2\ | 2 / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ | | / / 2\ / / 2\ | | / / 2\ | | | \/ polar_lift\1 - y / y *\/ polar_lift\1 - y / 1 - y 3*x *\/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / y *\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \\/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \\/ polar_lift\1 - y / / | | - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------ + -------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------- otherwise | | ________________________ ________________________ ________________________ ________________________ / ________________________ ________________________\ ________________________ ________________________ 2 / ________________________ ________________________\ 2 2 2 ________________________ / ________________________ ________________________\ 2 ________________________ / ________________________ ________________________\ | | / 2 / 2 / 2 / 2 | / 2 / 2 | ____________________ / 2 / 2 / ________________________ ________________________\ | / 2 / 2 | ____________________ / ________________________ ________________________\ / ________________________ ________________________\ / ________________________ ________________________\ / 2 | / 2 / 2 | / ________________________ ________________________\ / 2 | / 2 / 2 | | | / x 2 / x / x 2 / x | / x 2 / x | / / 2\ / x 2 / x | / 2 / 2 | | / x 2 / x | / / 2\ | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | / x | / x 2 / x | / 2\ | / 2 / 2 | / x | / x 2 / x | / 2\ | | - 2* / 1 - ------------------ + 2*y * / 1 - ------------------ - 2* / 1 - ------------------ + 2*y * / 1 - ------------------ |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ |*\/ polar_lift\1 - y / - 2* / 1 - ------------------ + 2*y * / 1 - ------------------ | / x 2 / x | |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ |*\/ polar_lift\1 - y / | / x 2 / x | | / x 2 / x | | / x 2 / x | / 1 - ------------------ *|- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ |*polar_lift\1 - y / | / x 2 / x | / 1 - ------------------ *|- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ |*polar_lift\1 - y / | | / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | / / 2\ / / 2\ |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ | | / / 2\ / / 2\ | |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ | / / 2\ | / / 2\ / / 2\ | |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ | / / 2\ | / / 2\ / / 2\ | | | \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \/ polar_lift\1 - y / \/ polar_lift\1 - y / | / / 2\ / / 2\ | \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / / 2\ / / 2\ | \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | \ \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{2 i x^{3} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \left(1 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{\pi x y^{2} \left(1 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{2 i x \left(1 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{\pi x \left(1 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{2 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{\pi y^{2} \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(1 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} - \frac{2 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} + \frac{2 i \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{\pi \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{2 i \left(1 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} + \frac{2 i \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{y^{2} - 1}\right|} > 1 \\\frac{x^{3} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{3 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{x y^{2} \left(1 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{x y^{2} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{x \left(1 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(1 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} + \frac{y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{\left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} - \frac{1 - y^{2}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} - \frac{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dx$$
=
=
1 / | | / _________________________ _________________________ _________________________ _________________________ | | / 2 / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | / x / 2\ | 4*x 4*x*y | 3 / / 2\ | 4*x 4*x*y | / / 2\ | 4*x 4*x*y | 2 / x / 2\ | 4*x 4*x*y | 2 / / 2\ | 4*x 4*x*y | / x / 2\ | 4*x 4*x*y | / x \ 2 / x / 2\ | 4*x 4*x*y | / x \ | | pi* / -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------| 2*I*x *\/ polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------| 2*I*x*\/ polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------| pi*y * / -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------| 2*I*x*y *\/ polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------| 2*I* / -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------| 2*I*y * / -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------| | | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | ____________________| / / 2\ | _________________________ _________________________ | | ____________________| | | \/ polar_lift\1 - y / | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - y / | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - y / | / 2 / 2 | | / / 2\ | \/ polar_lift\1 - y / | / 2 / 2 | | / / 2\ | / 2\ / x \ 2 / 2\ / x \ | | | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| \\/ polar_lift\1 - y / / | / x / 2\ / x / 2\| \\/ polar_lift\1 - y / / 2*I*x*\1 - y /*acosh|-----------------------| 2*I*x*y *\1 - y /*acosh|-----------------------| | | ____________________ ____________________ ____________________ | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | / -1 + ------------------ *polar_lift\1 - y / / -1 + ------------------ *polar_lift\1 - y /| | ____________________| | ____________________| | | / / 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 - y / 6*I*x *\/ polar_lift\1 - y / 2*I*y *\/ polar_lift\1 - y / 2*I*\1 - y / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / 2*I*y *\1 - y / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / pi*x*\1 - y / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / pi*x*y *\1 - y / \\/ polar_lift\1 - y / \/ polar_lift\1 - y / / \\/ polar_lift\1 - y / / \\/ polar_lift\1 - y / / x | |-------------------------------------------------------------------------- - -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- 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 + y | | | / x 2 / x / x 2 / x / x 2 / x | / x 2 / x | / / 2\ | / 2 / 2 | | / 2 / 2 | | / x 2 / x | / / 2\ | / 2 / 2 | | / 2 / 2 | / x | / x 2 / x | / 2\ | / 2 / 2 | | / 2 / 2 | / x | / x 2 / x | / 2\ | / 2 / 2 | / x | / x 2 / x | / 2\ / x | / x 2 / x | / 2\ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ - 4* / -1 + ------------------ + 4*y * / -1 + ------------------ - 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |*\/ polar_lift\1 - y / | / x 2 / x | | / x 2 / x | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |*\/ polar_lift\1 - y / | / x 2 / x | | / x 2 / x | / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\1 - y / | / x 2 / x | | / x 2 / x | / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\1 - y / | / x 2 / x | / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\1 - y / / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\1 - y / | | / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ | | | \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / / 2\ / / 2\ | \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | | \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | < dx | | ________________________ ________________________ | | ____________________ / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | 3 / / 2\ | 2*x 2*x*y | / / 2\ | 2*x 2*x*y | 2 / / 2\ | 2*x 2*x*y | / x / 2\ | 2*x 2*x*y | / x \ 2 / x / 2\ | 2*x 2*x*y | / x \ | | x *\/ polar_lift\1 - y / *|- ------------------------------------------------- + -------------------------------------------------| x*\/ polar_lift\1 - y / *|- ------------------------------------------------- + -------------------------------------------------| x*y *\/ polar_lift\1 - y / *|- ------------------------------------------------- + -------------------------------------------------| / 1 - ------------------ *\1 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------| y * / 1 - ------------------ *\1 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------| | | | ________________________ ________________________ | | ________________________ ________________________ | | ________________________ ________________________ | / / 2\ | ________________________ ________________________ | | ____________________| / / 2\ | ________________________ ________________________ | | ____________________| | | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - y / | / 2 / 2 | | / / 2\ | / 2\ / x \ \/ polar_lift\1 - y / | / 2 / 2 | | / / 2\ | 2 / 2\ / x \ | | | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| | / x / 2\ / x / 2\| \\/ polar_lift\1 - y / / x*\1 - y /*asin|-----------------------| | / x / 2\ / x / 2\| \\/ polar_lift\1 - y / / x*y *\1 - y /*asin|-----------------------| | | ____________________ ____________________ ____________________ | / 1 - ------------------ *polar_lift\1 - y / / 1 - ------------------ *polar_lift\1 - y /| | / 1 - ------------------ *polar_lift\1 - y / / 1 - ------------------ *polar_lift\1 - y /| | / 1 - ------------------ *polar_lift\1 - y / / 1 - ------------------ *polar_lift\1 - y /| | / 1 - ------------------ *polar_lift\1 - y / / 1 - ------------------ *polar_lift\1 - y /| | ____________________| | / 1 - ------------------ *polar_lift\1 - y / / 1 - ------------------ *polar_lift\1 - y /| | ____________________| | | / / 2\ 2 / / 2\ 2 2 / / 2\ | / / 2\ / / 2\ | 2 / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ | | / / 2\ / / 2\ | | / / 2\ | | | \/ polar_lift\1 - y / y *\/ polar_lift\1 - y / 1 - y 3*x *\/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / y *\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \\/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \\/ polar_lift\1 - y / / | | - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------ + -------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------- otherwise | | ________________________ ________________________ ________________________ ________________________ / ________________________ ________________________\ ________________________ ________________________ 2 / ________________________ ________________________\ 2 2 2 ________________________ / ________________________ ________________________\ 2 ________________________ / ________________________ ________________________\ | | / 2 / 2 / 2 / 2 | / 2 / 2 | ____________________ / 2 / 2 / ________________________ ________________________\ | / 2 / 2 | ____________________ / ________________________ ________________________\ / ________________________ ________________________\ / ________________________ ________________________\ / 2 | / 2 / 2 | / ________________________ ________________________\ / 2 | / 2 / 2 | | | / x 2 / x / x 2 / x | / x 2 / x | / / 2\ / x 2 / x | / 2 / 2 | | / x 2 / x | / / 2\ | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | / x | / x 2 / x | / 2\ | / 2 / 2 | / x | / x 2 / x | / 2\ | | - 2* / 1 - ------------------ + 2*y * / 1 - ------------------ - 2* / 1 - ------------------ + 2*y * / 1 - ------------------ |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ |*\/ polar_lift\1 - y / - 2* / 1 - ------------------ + 2*y * / 1 - ------------------ | / x 2 / x | |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ |*\/ polar_lift\1 - y / | / x 2 / x | | / x 2 / x | | / x 2 / x | / 1 - ------------------ *|- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ |*polar_lift\1 - y / | / x 2 / x | / 1 - ------------------ *|- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ |*polar_lift\1 - y / | | / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | / / 2\ / / 2\ |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ | | / / 2\ / / 2\ | |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ | / / 2\ | / / 2\ / / 2\ | |- 2* / 1 - ------------------ + 2*y * / 1 - ------------------ | / / 2\ | / / 2\ / / 2\ | | | \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \/ polar_lift\1 - y / \/ polar_lift\1 - y / | / / 2\ / / 2\ | \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / / 2\ / / 2\ | \/ polar_lift\1 - y / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | \ \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / \ \/ polar_lift\1 - y / \/ polar_lift\1 - y / / | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{2 i x^{3} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \left(1 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{\pi x y^{2} \left(1 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{2 i x \left(1 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{\pi x \left(1 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{2 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{\pi y^{2} \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(1 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} - \frac{2 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} + \frac{2 i \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{\pi \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{2 i \left(1 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} + \frac{2 i \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{y^{2} - 1}\right|} > 1 \\\frac{x^{3} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{3 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{x y^{2} \left(1 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{x y^{2} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{x \left(1 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(1 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} + \frac{y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{\left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} - \frac{1 - y^{2}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} - \frac{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((2*i*sqrt(polar_lift(1 - y^2))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2))) - 6*i*x^2*sqrt(polar_lift(1 - y^2))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2))) - 2*i*y^2*sqrt(polar_lift(1 - y^2))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2))) + 2*i*(1 - y^2)/((-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))*sqrt(polar_lift(1 - y^2))) - pi*sqrt(-1 + x^2/polar_lift(1 - y^2))*(1 - y^2)*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 - 2*i*x^3*sqrt(polar_lift(1 - y^2))*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 - 2*i*y^2*(1 - y^2)/((-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))*sqrt(polar_lift(1 - y^2))) + 2*i*x*sqrt(polar_lift(1 - y^2))*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 + pi*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2))*(1 - y^2)*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 - pi*x*(1 - y^2)/(sqrt(-1 + x^2/polar_lift(1 - y^2))*(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))*polar_lift(1 - y^2)) - 2*i*x*y^2*sqrt(polar_lift(1 - y^2))*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 + 2*i*sqrt(-1 + x^2/polar_lift(1 - y^2))*(1 - y^2)*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))*acosh(x/sqrt(polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 + pi*x*y^2*(1 - y^2)/(sqrt(-1 + x^2/polar_lift(1 - y^2))*(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))*polar_lift(1 - y^2)) - 2*i*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2))*(1 - y^2)*(4*x/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))*acosh(x/sqrt(polar_lift(1 - y^2)))/(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))^2 + 2*i*x*(1 - y^2)*acosh(x/sqrt(polar_lift(1 - y^2)))/(sqrt(-1 + x^2/polar_lift(1 - y^2))*(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))*polar_lift(1 - y^2)) - 2*i*x*y^2*(1 - y^2)*acosh(x/sqrt(polar_lift(1 - y^2)))/(sqrt(-1 + x^2/polar_lift(1 - y^2))*(-4*sqrt(-1 + x^2/polar_lift(1 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2)))*polar_lift(1 - y^2)), x^2/|-1 + y^2| > 1), (-sqrt(polar_lift(1 - y^2))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2))) + y^2*sqrt(polar_lift(1 - y^2))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2))) - (1 - y^2)/((-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))*sqrt(polar_lift(1 - y^2))) + 3*x^2*sqrt(polar_lift(1 - y^2))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2))) + x^3*sqrt(polar_lift(1 - y^2))*(-2*x/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))^2 + y^2*(1 - y^2)/((-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))*sqrt(polar_lift(1 - y^2))) - x*sqrt(polar_lift(1 - y^2))*(-2*x/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))^2 + x*y^2*sqrt(polar_lift(1 - y^2))*(-2*x/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))^2 - sqrt(1 - x^2/polar_lift(1 - y^2))*(1 - y^2)*(-2*x/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))*asin(x/sqrt(polar_lift(1 - y^2)))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))^2 + x*(1 - y^2)*asin(x/sqrt(polar_lift(1 - y^2)))/(sqrt(1 - x^2/polar_lift(1 - y^2))*(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))*polar_lift(1 - y^2)) + y^2*sqrt(1 - x^2/polar_lift(1 - y^2))*(1 - y^2)*(-2*x/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2))*polar_lift(1 - y^2)))*asin(x/sqrt(polar_lift(1 - y^2)))/(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))^2 - x*y^2*(1 - y^2)*asin(x/sqrt(polar_lift(1 - y^2)))/(sqrt(1 - x^2/polar_lift(1 - y^2))*(-2*sqrt(1 - x^2/polar_lift(1 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2)))*polar_lift(1 - y^2)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.