Sr Examen
Integral de x+sqrt(4-x^2-y^2)+y dx
Solución
Ha introducido
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1 / | | / _____________ \ | | / 2 2 | | \x + \/ 4 - x - y + y/ dx | / 0
$$\int\limits_{0}^{1} \left(y + \left(x + \sqrt{- y^{2} + \left(4 - x^{2}\right)}\right)\right)\, dx$$
Integral(x + sqrt(4 - x^2 - y^2) + y, (x, 0, 1))
Respuesta (Indefinida)
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// _________________________ _________________________ \
|| _________________________ _________________________ / 2 / 2 |
|| / 2 / 2 / x / 2\ / x \ 2 / x / 2\ / x \ |
|| / x / 2\ 2 / x / 2\ 8*I* / -1 + ------------------ *\4 - y /*acosh|-----------------------| 2*I*y * / -1 + ------------------ *\4 - y /*acosh|-----------------------| |
|| 4*pi* / -1 + ------------------ *\4 - y / ____________________ ____________________ pi*y * / -1 + ------------------ *\4 - y / ____________________ / / 2\ | ____________________| / / 2\ | ____________________| |
|| / / 2\ 3 / / 2\ / / 2\ / / 2\ 2 / / 2\ \/ polar_lift\4 - y / | / / 2\ | \/ polar_lift\4 - y / | / / 2\ | | 2 | |
|| \/ polar_lift\4 - y / 2*I*x *\/ polar_lift\4 - y / 8*I*x*\/ polar_lift\4 - y / \/ polar_lift\4 - y / 2*I*x*y *\/ polar_lift\4 - y / \\/ polar_lift\4 - y / / \\/ polar_lift\4 - y / / | x | |
||- --------------------------------------------------------------------------- - --------------------------------------------------------------------------- + --------------------------------------------------------------------------- + --------------------------------------------------------------------------- - --------------------------------------------------------------------------- + --------------------------------------------------------------------------- - ------------------------------------------------------------------------------ for |-------| > 1|
|| _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ | 2| |
|| / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 |-4 + y | |
/ || / x 2 / x / x 2 / x / x 2 / x / x 2 / x / x 2 / x / x 2 / x / x 2 / x |
| || - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |
| / _____________ \ 2 || / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ |
| | / 2 2 | x || \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / |
| \x + \/ 4 - x - y + y/ dx = C + -- + x*y + |< |
| 2 || ________________________ ________________________ |
/ || / 2 / 2 |
|| / x / 2\ / x \ 2 / x / 2\ / x \ |
|| 4* / 1 - ------------------ *\4 - y /*asin|-----------------------| y * / 1 - ------------------ *\4 - y /*asin|-----------------------| |
|| ____________________ ____________________ ____________________ / / 2\ | ____________________| / / 2\ | ____________________| |
|| 3 / / 2\ / / 2\ 2 / / 2\ \/ polar_lift\4 - y / | / / 2\ | \/ polar_lift\4 - y / | / / 2\ | |
|| x *\/ polar_lift\4 - y / 4*x*\/ polar_lift\4 - y / x*y *\/ polar_lift\4 - y / \\/ polar_lift\4 - y / / \\/ polar_lift\4 - 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 |
|| - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |
|| / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ |
\\ \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / /
$$\int \left(y + \left(x + \sqrt{- y^{2} + \left(4 - x^{2}\right)}\right)\right)\, dx = C + \frac{x^{2}}{2} + x y + \begin{cases} - \frac{2 i x^{3} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{8 i x \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{2 i y^{2} \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{\pi y^{2} \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{8 i \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{4 \pi \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2} - 4}}\right| > 1 \\\frac{x^{3} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} + \frac{x y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{4 x \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} + \frac{y^{2} \left(4 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{4 \left(4 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}$$
Respuesta
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1
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| / _________________________ _________________________ _________________________ _________________________
| | / 2 / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \
| | / x / 2\ | 16*x 4*x*y | 3 / / 2\ | 16*x 4*x*y | / / 2\ | 16*x 4*x*y | 2 / x / 2\ | 16*x 4*x*y | 2 / / 2\ | 16*x 4*x*y | / x / 2\ | 16*x 4*x*y | / x \ 2 / x / 2\ | 16*x 4*x*y | / x \
| | 4*pi* / -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------| 2*I*x *\/ polar_lift\4 - y / *|-------------------------------------------------- - --------------------------------------------------| 8*I*x*\/ polar_lift\4 - y / *|-------------------------------------------------- - --------------------------------------------------| pi*y * / -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------| 2*I*x*y *\/ polar_lift\4 - y / *|-------------------------------------------------- - --------------------------------------------------| 8*I* / -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------| 2*I*y * / -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|
| | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | ____________________| / / 2\ | _________________________ _________________________ | | ____________________|
| | \/ polar_lift\4 - y / | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\4 - y / | / 2 / 2 | | / 2 / 2 | \/ polar_lift\4 - y / | / 2 / 2 | | / / 2\ | \/ polar_lift\4 - 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\4 - y / / | / x / 2\ / x / 2\| \\/ polar_lift\4 - y / / 8*I*x*\4 - y /*acosh|-----------------------| 2*I*x*y *\4 - y /*acosh|-----------------------|
| | ____________________ ____________________ ____________________ | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - y /| | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - y /| | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - y /| | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - y /| | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - y /| | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - y /| | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - 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
| | 8*I*\/ polar_lift\4 - y / 6*I*x *\/ polar_lift\4 - y / 2*I*y *\/ polar_lift\4 - y / 8*I*\4 - y / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / 2*I*y *\4 - y / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / 4*pi*x*\4 - y / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / pi*x*y *\4 - y / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / \\/ polar_lift\4 - y / / \\/ polar_lift\4 - 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 | |-4 + 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\
| |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*\/ polar_lift\4 - y / | / x 2 / x | | / x 2 / x | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*\/ polar_lift\4 - y / | / x 2 / x | | / x 2 / x | / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\4 - y / | / x 2 / x | | / x 2 / x | / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\4 - y / | / x 2 / x | / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\4 - y / / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\4 - y /
| | / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | | / / 2\ / / 2\ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ |
| | \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / | / / 2\ / / 2\ | \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / /
1 | | \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / /
- + y + | < dx
2 | | ________________________ ________________________
| | ____________________ / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \
| | 3 / / 2\ | 8*x 2*x*y | / / 2\ | 8*x 2*x*y | 2 / / 2\ | 8*x 2*x*y | / x / 2\ | 8*x 2*x*y | / x \ 2 / x / 2\ | 8*x 2*x*y | / x \
| | x *\/ polar_lift\4 - y / *|- ------------------------------------------------- + -------------------------------------------------| 4*x*\/ polar_lift\4 - y / *|- ------------------------------------------------- + -------------------------------------------------| x*y *\/ polar_lift\4 - y / *|- ------------------------------------------------- + -------------------------------------------------| 4* / 1 - ------------------ *\4 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------| y * / 1 - ------------------ *\4 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|
| | | ________________________ ________________________ | | ________________________ ________________________ | | ________________________ ________________________ | / / 2\ | ________________________ ________________________ | | ____________________| / / 2\ | ________________________ ________________________ | | ____________________|
| | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\4 - y / | / 2 / 2 | | / / 2\ | \/ polar_lift\4 - 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\| \\/ polar_lift\4 - y / / | / x / 2\ / x / 2\| \\/ polar_lift\4 - y / / 4*x*\4 - y /*asin|-----------------------| x*y *\4 - y /*asin|-----------------------|
| | ____________________ ____________________ ____________________ | / 1 - ------------------ *polar_lift\4 - y / / 1 - ------------------ *polar_lift\4 - y /| | / 1 - ------------------ *polar_lift\4 - y / / 1 - ------------------ *polar_lift\4 - y /| | / 1 - ------------------ *polar_lift\4 - y / / 1 - ------------------ *polar_lift\4 - y /| | / 1 - ------------------ *polar_lift\4 - y / / 1 - ------------------ *polar_lift\4 - y /| | / 1 - ------------------ *polar_lift\4 - y / / 1 - ------------------ *polar_lift\4 - y /| | ____________________| | ____________________|
| | / / 2\ 2 / / 2\ / 2\ 2 / / 2\ | / / 2\ / / 2\ | 2 / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ | | / / 2\ |
| | 4*\/ polar_lift\4 - y / y *\/ polar_lift\4 - y / 4*\4 - y / 3*x *\/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / y *\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \\/ polar_lift\4 - y / / \\/ polar_lift\4 - 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 | | / 2 / 2 | / x | / x 2 / x | / 2\ / x | / x 2 / x | / 2\
| | - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |*\/ polar_lift\4 - y / - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | / x 2 / x | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |*\/ polar_lift\4 - y / | / x 2 / x | | / x 2 / x | | / x 2 / x | | / x 2 / x | / 1 - ------------------ *|- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |*polar_lift\4 - y / / 1 - ------------------ *|- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |*polar_lift\4 - y /
| | / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | / / 2\ / / 2\ |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | | / / 2\ / / 2\ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | / / 2\ | / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ |
| | \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \/ polar_lift\4 - y / \/ polar_lift\4 - y / | / / 2\ / / 2\ | \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / /
| \ \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / /
|
/
0
$$y + \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(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \left(4 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{\pi x y^{2} \left(4 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{2 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} + \frac{8 i x \left(4 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{4 \pi x \left(4 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{8 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} + \frac{\pi y^{2} \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(4 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} - \frac{2 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{8 i \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{4 \pi \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} + \frac{8 i \left(4 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} + \frac{8 i \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{y^{2} - 4}\right|} > 1 \\\frac{x^{3} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{3 x^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{x y^{2} \left(4 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{x y^{2} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{4 x \left(4 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{4 x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(4 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(4 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} + \frac{y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{4 \left(4 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} - \frac{4 \left(4 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} - \frac{4 \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dx + \frac{1}{2}$$
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| / _________________________ _________________________ _________________________ _________________________
| | / 2 / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \
| | / x / 2\ | 16*x 4*x*y | 3 / / 2\ | 16*x 4*x*y | / / 2\ | 16*x 4*x*y | 2 / x / 2\ | 16*x 4*x*y | 2 / / 2\ | 16*x 4*x*y | / x / 2\ | 16*x 4*x*y | / x \ 2 / x / 2\ | 16*x 4*x*y | / x \
| | 4*pi* / -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------| 2*I*x *\/ polar_lift\4 - y / *|-------------------------------------------------- - --------------------------------------------------| 8*I*x*\/ polar_lift\4 - y / *|-------------------------------------------------- - --------------------------------------------------| pi*y * / -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------| 2*I*x*y *\/ polar_lift\4 - y / *|-------------------------------------------------- - --------------------------------------------------| 8*I* / -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------| 2*I*y * / -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|
| | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | ____________________| / / 2\ | _________________________ _________________________ | | ____________________|
| | \/ polar_lift\4 - y / | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\4 - y / | / 2 / 2 | | / 2 / 2 | \/ polar_lift\4 - y / | / 2 / 2 | | / / 2\ | \/ polar_lift\4 - 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\4 - y / / | / x / 2\ / x / 2\| \\/ polar_lift\4 - y / / 8*I*x*\4 - y /*acosh|-----------------------| 2*I*x*y *\4 - y /*acosh|-----------------------|
| | ____________________ ____________________ ____________________ | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - y /| | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - y /| | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - y /| | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - y /| | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - y /| | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - y /| | / -1 + ------------------ *polar_lift\4 - y / / -1 + ------------------ *polar_lift\4 - 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
| | 8*I*\/ polar_lift\4 - y / 6*I*x *\/ polar_lift\4 - y / 2*I*y *\/ polar_lift\4 - y / 8*I*\4 - y / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / 2*I*y *\4 - y / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / 4*pi*x*\4 - y / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / pi*x*y *\4 - y / \\/ polar_lift\4 - y / \/ polar_lift\4 - y / / \\/ polar_lift\4 - y / / \\/ polar_lift\4 - 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 | |-4 + 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\
| |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*\/ polar_lift\4 - y / | / x 2 / x | | / x 2 / x | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*\/ polar_lift\4 - y / | / x 2 / x | | / x 2 / x | / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\4 - y / | / x 2 / x | | / x 2 / x | / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\4 - y / | / x 2 / x | / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\4 - y / / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift\4 - y /
| | / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | | / / 2\ / / 2\ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ |
| | \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / | / / 2\ / / 2\ | \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / /
1 | | \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / /
- + y + | < dx
2 | | ________________________ ________________________
| | ____________________ / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \
| | 3 / / 2\ | 8*x 2*x*y | / / 2\ | 8*x 2*x*y | 2 / / 2\ | 8*x 2*x*y | / x / 2\ | 8*x 2*x*y | / x \ 2 / x / 2\ | 8*x 2*x*y | / x \
| | x *\/ polar_lift\4 - y / *|- ------------------------------------------------- + -------------------------------------------------| 4*x*\/ polar_lift\4 - y / *|- ------------------------------------------------- + -------------------------------------------------| x*y *\/ polar_lift\4 - y / *|- ------------------------------------------------- + -------------------------------------------------| 4* / 1 - ------------------ *\4 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------| y * / 1 - ------------------ *\4 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|
| | | ________________________ ________________________ | | ________________________ ________________________ | | ________________________ ________________________ | / / 2\ | ________________________ ________________________ | | ____________________| / / 2\ | ________________________ ________________________ | | ____________________|
| | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\4 - y / | / 2 / 2 | | / / 2\ | \/ polar_lift\4 - 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\| \\/ polar_lift\4 - y / / | / x / 2\ / x / 2\| \\/ polar_lift\4 - y / / 4*x*\4 - y /*asin|-----------------------| x*y *\4 - y /*asin|-----------------------|
| | ____________________ ____________________ ____________________ | / 1 - ------------------ *polar_lift\4 - y / / 1 - ------------------ *polar_lift\4 - y /| | / 1 - ------------------ *polar_lift\4 - y / / 1 - ------------------ *polar_lift\4 - y /| | / 1 - ------------------ *polar_lift\4 - y / / 1 - ------------------ *polar_lift\4 - y /| | / 1 - ------------------ *polar_lift\4 - y / / 1 - ------------------ *polar_lift\4 - y /| | / 1 - ------------------ *polar_lift\4 - y / / 1 - ------------------ *polar_lift\4 - y /| | ____________________| | ____________________|
| | / / 2\ 2 / / 2\ / 2\ 2 / / 2\ | / / 2\ / / 2\ | 2 / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ | | / / 2\ |
| | 4*\/ polar_lift\4 - y / y *\/ polar_lift\4 - y / 4*\4 - y / 3*x *\/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / y *\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \\/ polar_lift\4 - y / / \\/ polar_lift\4 - 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 | | / 2 / 2 | / x | / x 2 / x | / 2\ / x | / x 2 / x | / 2\
| | - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |*\/ polar_lift\4 - y / - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | / x 2 / x | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |*\/ polar_lift\4 - y / | / x 2 / x | | / x 2 / x | | / x 2 / x | | / x 2 / x | / 1 - ------------------ *|- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |*polar_lift\4 - y / / 1 - ------------------ *|- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |*polar_lift\4 - y /
| | / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | / / 2\ / / 2\ |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | | / / 2\ / / 2\ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | / / 2\ | / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ |
| | \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \/ polar_lift\4 - y / \/ polar_lift\4 - y / | / / 2\ / / 2\ | \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \/ polar_lift\4 - y / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / /
| \ \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / / \ \/ polar_lift\4 - y / \/ polar_lift\4 - y / /
|
/
0
$$y + \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(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \left(4 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{\pi x y^{2} \left(4 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{2 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} + \frac{8 i x \left(4 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{4 \pi x \left(4 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{8 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} + \frac{\pi y^{2} \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(4 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} - \frac{2 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{8 i \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{4 \pi \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} + \frac{8 i \left(4 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} + \frac{8 i \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{y^{2} - 4}\right|} > 1 \\\frac{x^{3} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{3 x^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{x y^{2} \left(4 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{x y^{2} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{4 x \left(4 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{4 x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(4 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(4 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} + \frac{y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{4 \left(4 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} - \frac{4 \left(4 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} - \frac{4 \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dx + \frac{1}{2}$$
1/2 + y + Integral(Piecewise((8*i*sqrt(polar_lift(4 - y^2))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2))) - 6*i*x^2*sqrt(polar_lift(4 - y^2))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2))) - 2*i*y^2*sqrt(polar_lift(4 - y^2))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2))) + 8*i*(4 - y^2)/((-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))*sqrt(polar_lift(4 - y^2))) - 4*pi*sqrt(-1 + x^2/polar_lift(4 - y^2))*(4 - y^2)*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 - 2*i*x^3*sqrt(polar_lift(4 - y^2))*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 - 2*i*y^2*(4 - y^2)/((-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))*sqrt(polar_lift(4 - y^2))) + 8*i*x*sqrt(polar_lift(4 - y^2))*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 + pi*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2))*(4 - y^2)*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 - 4*pi*x*(4 - y^2)/(sqrt(-1 + x^2/polar_lift(4 - y^2))*(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))*polar_lift(4 - y^2)) - 2*i*x*y^2*sqrt(polar_lift(4 - y^2))*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 + 8*i*sqrt(-1 + x^2/polar_lift(4 - y^2))*(4 - y^2)*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))*acosh(x/sqrt(polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 + pi*x*y^2*(4 - y^2)/(sqrt(-1 + x^2/polar_lift(4 - y^2))*(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))*polar_lift(4 - y^2)) - 2*i*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2))*(4 - y^2)*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))*acosh(x/sqrt(polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 + 8*i*x*(4 - y^2)*acosh(x/sqrt(polar_lift(4 - y^2)))/(sqrt(-1 + x^2/polar_lift(4 - y^2))*(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))*polar_lift(4 - y^2)) - 2*i*x*y^2*(4 - y^2)*acosh(x/sqrt(polar_lift(4 - y^2)))/(sqrt(-1 + x^2/polar_lift(4 - y^2))*(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))*polar_lift(4 - y^2)), x^2/|-4 + y^2| > 1), (-4*sqrt(polar_lift(4 - y^2))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2))) + y^2*sqrt(polar_lift(4 - y^2))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2))) - 4*(4 - y^2)/((-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))*sqrt(polar_lift(4 - y^2))) + 3*x^2*sqrt(polar_lift(4 - y^2))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2))) + x^3*sqrt(polar_lift(4 - y^2))*(-8*x/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))^2 + y^2*(4 - y^2)/((-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))*sqrt(polar_lift(4 - y^2))) - 4*x*sqrt(polar_lift(4 - y^2))*(-8*x/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))^2 + x*y^2*sqrt(polar_lift(4 - y^2))*(-8*x/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))^2 - 4*sqrt(1 - x^2/polar_lift(4 - y^2))*(4 - y^2)*(-8*x/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))*asin(x/sqrt(polar_lift(4 - y^2)))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))^2 + y^2*sqrt(1 - x^2/polar_lift(4 - y^2))*(4 - y^2)*(-8*x/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))*asin(x/sqrt(polar_lift(4 - y^2)))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))^2 + 4*x*(4 - y^2)*asin(x/sqrt(polar_lift(4 - y^2)))/(sqrt(1 - x^2/polar_lift(4 - y^2))*(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))*polar_lift(4 - y^2)) - x*y^2*(4 - y^2)*asin(x/sqrt(polar_lift(4 - y^2)))/(sqrt(1 - x^2/polar_lift(4 - y^2))*(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))*polar_lift(4 - y^2)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.