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