Sr Examen
Integral de 3*sqrt(1-x^2-y^2/4) dx
Solución
Ha introducido
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1 / | | _____________ | / 2 | / 2 y | 3* / 1 - x - -- dx | \/ 4 | / 0
$$\int\limits_{0}^{1} 3 \sqrt{- \frac{y^{2}}{4} + \left(1 - x^{2}\right)}\, dx$$
Integral(3*sqrt(1 - x^2 - y^2/4), (x, 0, 1))
Solución detallada
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La integral del producto de una función por una constante es la constante por la integral de esta función:
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Hay varias maneras de calcular esta integral.
Método #1
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Vuelva a escribir el integrando:
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La integral del producto de una función por una constante es la constante por la integral de esta función:
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No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
Por lo tanto, el resultado es:
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Método #2
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Vuelva a escribir el integrando:
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Vuelva a escribir el integrando:
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La integral del producto de una función por una constante es la constante por la integral de esta función:
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No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
Por lo tanto, el resultado es:
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Por lo tanto, el resultado es:
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Ahora simplificar:
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Añadimos la constante de integración:
Respuesta:
Respuesta (Indefinida)
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// _________________________ _________________________ \
|| _________________________ _________________________ / 2 / 2 |
|| / 2 / 2 / 4*x / 2\ / 2*x \ 2 / 4*x / 2\ / 2*x \ |
|| / 4*x / 2\ 2 / 4*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\ | ____________________| |
|| 3 / / 2\ / / 2\ / / 2\ / / 2\ 2 / / 2\ \/ polar_lift\4 - y / | / / 2\ | \/ polar_lift\4 - y / | / / 2\ | | 2 | |
|| 16*I*x *\/ polar_lift\4 - y / \/ polar_lift\4 - y / 16*I*x*\/ polar_lift\4 - y / \/ polar_lift\4 - y / 4*I*x*y *\/ polar_lift\4 - y / \\/ polar_lift\4 - y / / \\/ polar_lift\4 - y / / | x | |
||- --------------------------------------------------------------------------- - --------------------------------------------------------------------------- + --------------------------------------------------------------------------- + --------------------------------------------------------------------------- - --------------------------------------------------------------------------- + --------------------------------------------------------------------------- - ------------------------------------------------------------------------------ for 4*|-------| > 1|
|| _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ | 2| |
|| / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 |-4 + y | |
|| / 4*x 2 / 4*x / 4*x 2 / 4*x / 4*x 2 / 4*x / 4*x 2 / 4*x / 4*x 2 / 4*x / 4*x 2 / 4*x / 4*x 2 / 4*x |
|| - 32* / -1 + ------------------ + 8*y * / -1 + ------------------ - 32* / -1 + ------------------ + 8*y * / -1 + ------------------ - 32* / -1 + ------------------ + 8*y * / -1 + ------------------ - 32* / -1 + ------------------ + 8*y * / -1 + ------------------ - 32* / -1 + ------------------ + 8*y * / -1 + ------------------ - 32* / -1 + ------------------ + 8*y * / -1 + ------------------ - 32* / -1 + ------------------ + 8*y * / -1 + ------------------ |
|| / / 2\ / / 2\ / / 2\ / / 2\ / / 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 / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / \/ polar_lift\4 - y / |
3*|< |
|| ________________________ ________________________ |
|| / 2 / 2 |
|| / 4*x / 2\ / 2*x \ 2 / 4*x / 2\ / 2*x \ |
|| 4* / 1 - ------------------ *\4 - y /*asin|-----------------------| y * / 1 - ------------------ *\4 - y /*asin|-----------------------| |
|| ____________________ ____________________ / / 2\ | ____________________| ____________________ / / 2\ | ____________________| |
|| / / 2\ 3 / / 2\ \/ polar_lift\4 - y / | / / 2\ | 2 / / 2\ \/ polar_lift\4 - y / | / / 2\ | |
|| 8*x*\/ polar_lift\4 - y / 8*x *\/ polar_lift\4 - y / \\/ polar_lift\4 - y / / 2*x*y *\/ polar_lift\4 - y / \\/ polar_lift\4 - y / / |
|| - ------------------------------------------------------------------------- + ------------------------------------------------------------------------- - ------------------------------------------------------------------------- + ------------------------------------------------------------------------- + ------------------------------------------------------------------------- otherwise |
|| ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ |
/ || / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 |
| || / 4*x 2 / 4*x / 4*x 2 / 4*x / 4*x 2 / 4*x / 4*x 2 / 4*x / 4*x 2 / 4*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 - ------------------ |
| / 2 || / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ |
| / 2 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 / /
| 3* / 1 - x - -- dx = C + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
| \/ 4 2
|
/
$$\int 3 \sqrt{- \frac{y^{2}}{4} + \left(1 - x^{2}\right)}\, dx = C + \frac{3 \left(\begin{cases} - \frac{16 i x^{3} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{8 y^{2} \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 32 \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{4 i x y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{8 y^{2} \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 32 \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{16 i x \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{8 y^{2} \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 32 \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{2 i y^{2} \left(4 - y^{2}\right) \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{acosh}{\left(\frac{2 x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{8 y^{2} \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 32 \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{\pi y^{2} \left(4 - y^{2}\right) \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}}{8 y^{2} \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 32 \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{8 i \left(4 - y^{2}\right) \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{acosh}{\left(\frac{2 x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{8 y^{2} \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 32 \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{4 \pi \left(4 - y^{2}\right) \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}}{8 y^{2} \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 32 \sqrt{\frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} & \text{for}\: 4 \left|{\frac{x^{2}}{y^{2} - 4}}\right| > 1 \\\frac{8 x^{3} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{- \frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 16 \sqrt{- \frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} + \frac{2 x y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{- \frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 16 \sqrt{- \frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{8 x \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{- \frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 16 \sqrt{- \frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} + \frac{y^{2} \left(4 - y^{2}\right) \sqrt{- \frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{asin}{\left(\frac{2 x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{4 y^{2} \sqrt{- \frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 16 \sqrt{- \frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{4 \left(4 - y^{2}\right) \sqrt{- \frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{asin}{\left(\frac{2 x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{4 y^{2} \sqrt{- \frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 16 \sqrt{- \frac{4 x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\right)}{2}$$
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 / | y | | 16*x 4*x*y | / x | y | | 16*x 4*x*y | / | y | | 16*x 4*x*y | 2 / | y | | 16*x 4*x*y | 2 / x | y | | 16*x 4*x*y | / x | y | | 16*x 4*x*y | / x \ 2 / x | y | | 16*x 4*x*y | / x \ | | 24*I*x * / polar_lift|1 - --| *|---------------------------------------------------- - ----------------------------------------------------| 12*pi* / -1 + ------------------ *|1 - --|*|---------------------------------------------------- - ----------------------------------------------------| 24*I*x* / polar_lift|1 - --| *|---------------------------------------------------- - ----------------------------------------------------| 6*I*x*y * / polar_lift|1 - --| *|---------------------------------------------------- - ----------------------------------------------------| 3*pi*y * / -1 + ------------------ *|1 - --|*|---------------------------------------------------- - ----------------------------------------------------| 24*I* / -1 + ------------------ *|1 - --|*|---------------------------------------------------- - ----------------------------------------------------|*acosh|-------------------------| 6*I*y * / -1 + ------------------ *|1 - --|*|---------------------------------------------------- - ----------------------------------------------------|*acosh|-------------------------| | | \/ \ 4 / | _________________________ _________________________ | / / 2\ \ 4 / | _________________________ _________________________ | \/ \ 4 / | _________________________ _________________________ | \/ \ 4 / | _________________________ _________________________ | / / 2\ \ 4 / | _________________________ _________________________ | / / 2\ \ 4 / | _________________________ _________________________ | | ____________________| / / 2\ \ 4 / | _________________________ _________________________ | | ____________________| / 2\ / 2\ | | | / 2 / 2\ / 2 / 2\| / | y | | / 2 / 2\ / 2 / 2\| | / 2 / 2\ / 2 / 2\| | / 2 / 2\ / 2 / 2\| / | y | | / 2 / 2\ / 2 / 2\| / | y | | / 2 / 2\ / 2 / 2\| | / / 2\ | / | y | | / 2 / 2\ / 2 / 2\| | / / 2\ | | y | / x \ 2 | y | / x \ | | | / x | y | / x | y || / polar_lift|1 - --| | / x | y | / x | y || | / x | y | / x | y || | / x | y | / x | y || / polar_lift|1 - --| | / x | y | / x | y || / polar_lift|1 - --| | / x | y | / x | y || | / | y | | / polar_lift|1 - --| | / x | y | / x | y || | / | y | | 24*I*x*|1 - --|*acosh|-------------------------| 6*I*x*y *|1 - --|*acosh|-------------------------| | | ____________________ ____________________ ____________________ | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| \/ \ 4 / | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| \/ \ 4 / | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| \/ \ 4 / | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| | / polar_lift|1 - --| | \/ \ 4 / | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| | / polar_lift|1 - --| | \ 4 / | ____________________| \ 4 / | ____________________| | | / / 2\ / / 2\ / / 2\ / 2\ | / / 2\ \ 4 / / / 2\ \ 4 /| | / / 2\ \ 4 / / / 2\ \ 4 /| / 2\ | / / 2\ \ 4 / / / 2\ \ 4 /| / 2\ | / / 2\ \ 4 / / / 2\ \ 4 /| | / / 2\ \ 4 / / / 2\ \ 4 /| | / / 2\ \ 4 / / / 2\ \ 4 /| \\/ \ 4 / / | / / 2\ \ 4 / / / 2\ \ 4 /| \\/ \ 4 / / / 2\ | / / 2\ | | / / 2\ | | | / | y | 2 / | y | 2 / | y | | y | | / | y | / | y | | | / | y | / | y | | 2 | y | | / | y | / | y | | | y | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | 2 | y | | / | y | | | / | y | | | | 24*I* / polar_lift|1 - --| 72*I*x * / polar_lift|1 - --| 6*I*y * / polar_lift|1 - --| 24*I*|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | 6*I*y *|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | 12*pi*x*|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | 3*pi*x*y *|1 - --| | / polar_lift|1 - --| | | / polar_lift|1 - --| | 2 | | \/ \ 4 / \/ \ 4 / \/ \ 4 / \ 4 / \\/ \ 4 / \/ \ 4 / / \\/ \ 4 / \/ \ 4 / / \ 4 / \\/ \ 4 / \/ \ 4 / / \ 4 / \\/ \ 4 / \/ \ 4 / / \\/ \ 4 / \/ \ 4 / / \\/ \ 4 / \/ \ 4 / / \\/ \ 4 / \/ \ 4 / / \ 4 / \\/ \ 4 / / \\/ \ 4 / / x | |------------------------------------------------------------------------------- - ------------------------------------------------------------------------------- - ------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------- - 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-------------------------------------------------------------------------------------------------------------------------------------- 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 | / 2\ / 2 | / 2 / 2 | / 2\ | y | | | / x 2 / x / x 2 / x / x 2 / x | / x 2 / x | / | y | | / 2 / 2 | | / 2 / 2 | | / x 2 / x | / | y | | / 2 / 2 | / x | / x 2 / x | | y | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | / x | / x 2 / x | | y | / x | / x 2 / x | | y | / x | / x 2 / x | | y | |-1 + --| | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |* / polar_lift|1 - --| | / x 2 / x | | / x 2 / x | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |* / polar_lift|1 - --| | / x 2 / x | / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift|1 - --| | / x 2 / x | | / x 2 / x | | / x 2 / x | | / x 2 / x | / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift|1 - --| / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift|1 - --| / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift|1 - --| | 4 | | | / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | \/ \ 4 / |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | | / / 2\ / / 2\ | \/ \ 4 / |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | \ 4 / |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | \ 4 / / / 2\ | / / 2\ / / 2\ | \ 4 / / / 2\ | / / 2\ / / 2\ | \ 4 / | | / | y | / | y | / | y | / | y | / | y | / | y | | / | y | / | y | | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / | y | / | y | | | / / 2\ / / 2\ | / | y | | / | y | / | y | | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | / | y | | / | y | / | y | | / | y | | / | y | / | y | | / | y | | / | y | / | y | | | | / polar_lift|1 - --| / polar_lift|1 - --| / polar_lift|1 - --| / polar_lift|1 - --| / polar_lift|1 - --| / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | / | y | / | y | | | / | y | / | y | | | / polar_lift|1 - --| / polar_lift|1 - --| | | / | y | / | y | | / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | | \/ \ 4 / \/ \ 4 / \/ \ 4 / \/ \ 4 / \/ \ 4 / \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | \ \/ \ 4 / \/ \ 4 / / | / polar_lift|1 - --| / polar_lift|1 - --| | \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / | | \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / | < dx | | ____________________ ____________________ ________________________ ____________________ ________________________ | | / / 2\ / 2 \ / / 2\ / 2 \ / 2 / 2\ / 2 \ / / 2\ / 2 \ / 2 / 2\ / 2 \ | | / | y | | 8*x 2*x*y | 3 / | y | | 8*x 2*x*y | / x | y | | 8*x 2*x*y | / x \ 2 / | y | | 8*x 2*x*y | 2 / x | y | | 8*x 2*x*y | / x \ | | 12*x* / polar_lift|1 - --| *|- --------------------------------------------------- + ---------------------------------------------------| 12*x * / polar_lift|1 - --| *|- --------------------------------------------------- + ---------------------------------------------------| 12* / 1 - ------------------ *|1 - --|*|- --------------------------------------------------- + ---------------------------------------------------|*asin|-------------------------| 3*x*y * / polar_lift|1 - --| *|- --------------------------------------------------- + ---------------------------------------------------| 3*y * / 1 - ------------------ *|1 - --|*|- --------------------------------------------------- + ---------------------------------------------------|*asin|-------------------------| | | \/ \ 4 / | ________________________ ________________________ | \/ \ 4 / | ________________________ ________________________ | / / 2\ \ 4 / | ________________________ ________________________ | | ____________________| \/ \ 4 / | ________________________ ________________________ | / / 2\ \ 4 / | ________________________ ________________________ | | ____________________| / 2\ / 2\ | | | / 2 / 2\ / 2 / 2\| | / 2 / 2\ / 2 / 2\| / | y | | / 2 / 2\ / 2 / 2\| | / / 2\ | | / 2 / 2\ / 2 / 2\| / | y | | / 2 / 2\ / 2 / 2\| | / / 2\ | | y | / x \ 2 | y | / x \ | | | / x | y | / x | y || | / x | y | / x | y || / polar_lift|1 - --| | / x | y | / x | y || | / | y | | | / x | y | / x | y || / polar_lift|1 - --| | / x | y | / x | y || | / | y | | 12*x*|1 - --|*asin|-------------------------| 3*x*y *|1 - --|*asin|-------------------------| | | ____________________ ____________________ ____________________ | / 1 - ------------------ *polar_lift|1 - --| / 1 - ------------------ *polar_lift|1 - --|| | / 1 - ------------------ *polar_lift|1 - --| / 1 - ------------------ *polar_lift|1 - --|| \/ \ 4 / | / 1 - ------------------ *polar_lift|1 - --| / 1 - ------------------ *polar_lift|1 - --|| | / polar_lift|1 - --| | | / 1 - ------------------ *polar_lift|1 - --| / 1 - ------------------ *polar_lift|1 - --|| \/ \ 4 / | / 1 - ------------------ *polar_lift|1 - --| / 1 - ------------------ *polar_lift|1 - --|| | / polar_lift|1 - --| | \ 4 / | ____________________| \ 4 / | ____________________| | | / / 2\ / 2\ / / 2\ / / 2\ | / / 2\ \ 4 / / / 2\ \ 4 /| / 2\ | / / 2\ \ 4 / / / 2\ \ 4 /| | / / 2\ \ 4 / / / 2\ \ 4 /| \\/ \ 4 / / | / / 2\ \ 4 / / / 2\ \ 4 /| | / / 2\ \ 4 / / / 2\ \ 4 /| \\/ \ 4 / / | / / 2\ | | / / 2\ | | | / | y | | y | 2 / | y | 2 / | y | | / | y | / | y | | 2 | y | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | | / | y | | | / | y | | | | 12* / polar_lift|1 - --| 12*|1 - --| 3*y * / polar_lift|1 - --| 36*x * / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | 3*y *|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| | | / polar_lift|1 - --| | | | \/ \ 4 / \ 4 / \/ \ 4 / \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / \ 4 / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \\/ \ 4 / / \\/ \ 4 / / | | - ---------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------- + ---------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + ---------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------- otherwise | | ________________________ ________________________ / ________________________ ________________________\ ____________________ ________________________ ________________________ ________________________ ________________________ 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\ | | / x 2 / x | / x 2 / x | / | y | / x 2 / x / x 2 / x | / 2 / 2 | | / x 2 / x | / | y | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | / x | / x 2 / x | | y | / x | / x 2 / x | | y | | | - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |* / polar_lift|1 - --| - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | / x 2 / x | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |* / polar_lift|1 - --| | / x 2 / x | | / x 2 / x | | / x 2 / x | | / x 2 / x | / 1 - ------------------ *|- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |*polar_lift|1 - --| / 1 - ------------------ *|- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |*polar_lift|1 - --| | | / / 2\ / / 2\ | / / 2\ / / 2\ | \/ \ 4 / / / 2\ / / 2\ / / 2\ / / 2\ |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | | / / 2\ / / 2\ | \/ \ 4 / |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | / / 2\ | / / 2\ / / 2\ | \ 4 / / / 2\ | / / 2\ / / 2\ | \ 4 / | | / | y | / | y | | / | y | / | y | | / | y | / | y | / | y | / | y | | / / 2\ / / 2\ | | / | y | / | y | | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | / | y | | / | y | / | y | | / | y | | / | y | / | y | | | | / polar_lift|1 - --| / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| / polar_lift|1 - --| / polar_lift|1 - --| | / | y | / | y | | | / polar_lift|1 - --| / polar_lift|1 - --| | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | | \/ \ 4 / \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / \/ \ 4 / \/ \ 4 / \/ \ 4 / \/ \ 4 / | / polar_lift|1 - --| / polar_lift|1 - --| | \ \/ \ 4 / \/ \ 4 / / | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / | \ \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{24 i x^{3} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} - \frac{72 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}} - \frac{6 i x y^{2} \left(1 - \frac{y^{2}}{4}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{3 \pi x y^{2} \left(1 - \frac{y^{2}}{4}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{6 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} + \frac{24 i x \left(1 - \frac{y^{2}}{4}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{12 \pi x \left(1 - \frac{y^{2}}{4}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{24 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} - \frac{6 i y^{2} \left(1 - \frac{y^{2}}{4}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} + \frac{3 \pi y^{2} \left(1 - \frac{y^{2}}{4}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} - \frac{6 i y^{2} \left(1 - \frac{y^{2}}{4}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} - \frac{6 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}} + \frac{24 i \left(1 - \frac{y^{2}}{4}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} - \frac{12 \pi \left(1 - \frac{y^{2}}{4}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} + \frac{24 i \left(1 - \frac{y^{2}}{4}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} + \frac{24 i \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{\frac{y^{2}}{4} - 1}\right|} > 1 \\\frac{12 x^{3} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right)^{2}} + \frac{36 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}} - \frac{3 x y^{2} \left(1 - \frac{y^{2}}{4}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{3 x y^{2} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right)^{2}} + \frac{12 x \left(1 - \frac{y^{2}}{4}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{12 x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right)^{2}} + \frac{3 y^{2} \left(1 - \frac{y^{2}}{4}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right)^{2}} + \frac{3 y^{2} \left(1 - \frac{y^{2}}{4}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} + \frac{3 y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}} - \frac{12 \left(1 - \frac{y^{2}}{4}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right)^{2}} - \frac{12 \left(1 - \frac{y^{2}}{4}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} - \frac{12 \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \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 / | y | | 16*x 4*x*y | / x | y | | 16*x 4*x*y | / | y | | 16*x 4*x*y | 2 / | y | | 16*x 4*x*y | 2 / x | y | | 16*x 4*x*y | / x | y | | 16*x 4*x*y | / x \ 2 / x | y | | 16*x 4*x*y | / x \ | | 24*I*x * / polar_lift|1 - --| *|---------------------------------------------------- - ----------------------------------------------------| 12*pi* / -1 + ------------------ *|1 - --|*|---------------------------------------------------- - ----------------------------------------------------| 24*I*x* / polar_lift|1 - --| *|---------------------------------------------------- - ----------------------------------------------------| 6*I*x*y * / polar_lift|1 - --| *|---------------------------------------------------- - ----------------------------------------------------| 3*pi*y * / -1 + ------------------ *|1 - --|*|---------------------------------------------------- - ----------------------------------------------------| 24*I* / -1 + ------------------ *|1 - --|*|---------------------------------------------------- - ----------------------------------------------------|*acosh|-------------------------| 6*I*y * / -1 + ------------------ *|1 - --|*|---------------------------------------------------- - ----------------------------------------------------|*acosh|-------------------------| | | \/ \ 4 / | _________________________ _________________________ | / / 2\ \ 4 / | _________________________ _________________________ | \/ \ 4 / | _________________________ _________________________ | \/ \ 4 / | _________________________ _________________________ | / / 2\ \ 4 / | _________________________ _________________________ | / / 2\ \ 4 / | _________________________ _________________________ | | ____________________| / / 2\ \ 4 / | _________________________ _________________________ | | ____________________| / 2\ / 2\ | | | / 2 / 2\ / 2 / 2\| / | y | | / 2 / 2\ / 2 / 2\| | / 2 / 2\ / 2 / 2\| | / 2 / 2\ / 2 / 2\| / | y | | / 2 / 2\ / 2 / 2\| / | y | | / 2 / 2\ / 2 / 2\| | / / 2\ | / | y | | / 2 / 2\ / 2 / 2\| | / / 2\ | | y | / x \ 2 | y | / x \ | | | / x | y | / x | y || / polar_lift|1 - --| | / x | y | / x | y || | / x | y | / x | y || | / x | y | / x | y || / polar_lift|1 - --| | / x | y | / x | y || / polar_lift|1 - --| | / x | y | / x | y || | / | y | | / polar_lift|1 - --| | / x | y | / x | y || | / | y | | 24*I*x*|1 - --|*acosh|-------------------------| 6*I*x*y *|1 - --|*acosh|-------------------------| | | ____________________ ____________________ ____________________ | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| \/ \ 4 / | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| \/ \ 4 / | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| \/ \ 4 / | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| | / polar_lift|1 - --| | \/ \ 4 / | / -1 + ------------------ *polar_lift|1 - --| / -1 + ------------------ *polar_lift|1 - --|| | / polar_lift|1 - --| | \ 4 / | ____________________| \ 4 / | ____________________| | | / / 2\ / / 2\ / / 2\ / 2\ | / / 2\ \ 4 / / / 2\ \ 4 /| | / / 2\ \ 4 / / / 2\ \ 4 /| / 2\ | / / 2\ \ 4 / / / 2\ \ 4 /| / 2\ | / / 2\ \ 4 / / / 2\ \ 4 /| | / / 2\ \ 4 / / / 2\ \ 4 /| | / / 2\ \ 4 / / / 2\ \ 4 /| \\/ \ 4 / / | / / 2\ \ 4 / / / 2\ \ 4 /| \\/ \ 4 / / / 2\ | / / 2\ | | / / 2\ | | | / | y | 2 / | y | 2 / | y | | y | | / | y | / | y | | | / | y | / | y | | 2 | y | | / | y | / | y | | | y | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | 2 | y | | / | y | | | / | y | | | | 24*I* / polar_lift|1 - --| 72*I*x * / polar_lift|1 - --| 6*I*y * / polar_lift|1 - --| 24*I*|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | 6*I*y *|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | 12*pi*x*|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | 3*pi*x*y *|1 - --| | / polar_lift|1 - --| | | / polar_lift|1 - --| | 2 | | \/ \ 4 / \/ \ 4 / \/ \ 4 / \ 4 / \\/ \ 4 / \/ \ 4 / / \\/ \ 4 / \/ \ 4 / / \ 4 / \\/ \ 4 / \/ \ 4 / / \ 4 / \\/ \ 4 / \/ \ 4 / / \\/ \ 4 / \/ \ 4 / / \\/ \ 4 / \/ \ 4 / / \\/ \ 4 / \/ \ 4 / / \ 4 / \\/ \ 4 / / \\/ \ 4 / / 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 | / 2\ / 2 | / 2 / 2 | / 2\ | y | | | / x 2 / x / x 2 / x / x 2 / x | / x 2 / x | / | y | | / 2 / 2 | | / 2 / 2 | | / x 2 / x | / | y | | / 2 / 2 | / x | / x 2 / x | | y | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | / x | / x 2 / x | | y | / x | / x 2 / x | | y | / x | / x 2 / x | | y | |-1 + --| | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ - 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |* / polar_lift|1 - --| | / x 2 / x | | / x 2 / x | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |* / polar_lift|1 - --| | / x 2 / x | / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift|1 - --| | / x 2 / x | | / x 2 / x | | / x 2 / x | | / x 2 / x | / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift|1 - --| / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift|1 - --| / -1 + ------------------ *|- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ |*polar_lift|1 - --| | 4 | | | / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | \/ \ 4 / |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | | / / 2\ / / 2\ | \/ \ 4 / |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | \ 4 / |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | |- 16* / -1 + ------------------ + 4*y * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | \ 4 / / / 2\ | / / 2\ / / 2\ | \ 4 / / / 2\ | / / 2\ / / 2\ | \ 4 / | | / | y | / | y | / | y | / | y | / | y | / | y | | / | y | / | y | | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / | y | / | y | | | / / 2\ / / 2\ | / | y | | / | y | / | y | | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | / | y | | / | y | / | y | | / | y | | / | y | / | y | | / | y | | / | y | / | y | | | | / polar_lift|1 - --| / polar_lift|1 - --| / polar_lift|1 - --| / polar_lift|1 - --| / polar_lift|1 - --| / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | / | y | / | y | | | / | y | / | y | | | / polar_lift|1 - --| / polar_lift|1 - --| | | / | y | / | y | | / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | | \/ \ 4 / \/ \ 4 / \/ \ 4 / \/ \ 4 / \/ \ 4 / \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | \ \/ \ 4 / \/ \ 4 / / | / polar_lift|1 - --| / polar_lift|1 - --| | \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / | | \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / | < dx | | ____________________ ____________________ ________________________ ____________________ ________________________ | | / / 2\ / 2 \ / / 2\ / 2 \ / 2 / 2\ / 2 \ / / 2\ / 2 \ / 2 / 2\ / 2 \ | | / | y | | 8*x 2*x*y | 3 / | y | | 8*x 2*x*y | / x | y | | 8*x 2*x*y | / x \ 2 / | y | | 8*x 2*x*y | 2 / x | y | | 8*x 2*x*y | / x \ | | 12*x* / polar_lift|1 - --| *|- --------------------------------------------------- + ---------------------------------------------------| 12*x * / polar_lift|1 - --| *|- --------------------------------------------------- + ---------------------------------------------------| 12* / 1 - ------------------ *|1 - --|*|- --------------------------------------------------- + ---------------------------------------------------|*asin|-------------------------| 3*x*y * / polar_lift|1 - --| *|- --------------------------------------------------- + ---------------------------------------------------| 3*y * / 1 - ------------------ *|1 - --|*|- --------------------------------------------------- + ---------------------------------------------------|*asin|-------------------------| | | \/ \ 4 / | ________________________ ________________________ | \/ \ 4 / | ________________________ ________________________ | / / 2\ \ 4 / | ________________________ ________________________ | | ____________________| \/ \ 4 / | ________________________ ________________________ | / / 2\ \ 4 / | ________________________ ________________________ | | ____________________| / 2\ / 2\ | | | / 2 / 2\ / 2 / 2\| | / 2 / 2\ / 2 / 2\| / | y | | / 2 / 2\ / 2 / 2\| | / / 2\ | | / 2 / 2\ / 2 / 2\| / | y | | / 2 / 2\ / 2 / 2\| | / / 2\ | | y | / x \ 2 | y | / x \ | | | / x | y | / x | y || | / x | y | / x | y || / polar_lift|1 - --| | / x | y | / x | y || | / | y | | | / x | y | / x | y || / polar_lift|1 - --| | / x | y | / x | y || | / | y | | 12*x*|1 - --|*asin|-------------------------| 3*x*y *|1 - --|*asin|-------------------------| | | ____________________ ____________________ ____________________ | / 1 - ------------------ *polar_lift|1 - --| / 1 - ------------------ *polar_lift|1 - --|| | / 1 - ------------------ *polar_lift|1 - --| / 1 - ------------------ *polar_lift|1 - --|| \/ \ 4 / | / 1 - ------------------ *polar_lift|1 - --| / 1 - ------------------ *polar_lift|1 - --|| | / polar_lift|1 - --| | | / 1 - ------------------ *polar_lift|1 - --| / 1 - ------------------ *polar_lift|1 - --|| \/ \ 4 / | / 1 - ------------------ *polar_lift|1 - --| / 1 - ------------------ *polar_lift|1 - --|| | / polar_lift|1 - --| | \ 4 / | ____________________| \ 4 / | ____________________| | | / / 2\ / 2\ / / 2\ / / 2\ | / / 2\ \ 4 / / / 2\ \ 4 /| / 2\ | / / 2\ \ 4 / / / 2\ \ 4 /| | / / 2\ \ 4 / / / 2\ \ 4 /| \\/ \ 4 / / | / / 2\ \ 4 / / / 2\ \ 4 /| | / / 2\ \ 4 / / / 2\ \ 4 /| \\/ \ 4 / / | / / 2\ | | / / 2\ | | | / | y | | y | 2 / | y | 2 / | y | | / | y | / | y | | 2 | y | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | | / | y | | | / | y | | | | 12* / polar_lift|1 - --| 12*|1 - --| 3*y * / polar_lift|1 - --| 36*x * / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | 3*y *|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| | | / polar_lift|1 - --| | | | \/ \ 4 / \ 4 / \/ \ 4 / \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / \ 4 / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \\/ \ 4 / / \\/ \ 4 / / | | - ---------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------- + ---------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + ---------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------- otherwise | | ________________________ ________________________ / ________________________ ________________________\ ____________________ ________________________ ________________________ ________________________ ________________________ 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\ | | / x 2 / x | / x 2 / x | / | y | / x 2 / x / x 2 / x | / 2 / 2 | | / x 2 / x | / | y | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | / x | / x 2 / x | | y | / x | / x 2 / x | | y | | | - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |* / polar_lift|1 - --| - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ - 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | / x 2 / x | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |* / polar_lift|1 - --| | / x 2 / x | | / x 2 / x | | / x 2 / x | | / x 2 / x | / 1 - ------------------ *|- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |*polar_lift|1 - --| / 1 - ------------------ *|- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ |*polar_lift|1 - --| | | / / 2\ / / 2\ | / / 2\ / / 2\ | \/ \ 4 / / / 2\ / / 2\ / / 2\ / / 2\ |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | | / / 2\ / / 2\ | \/ \ 4 / |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | |- 8* / 1 - ------------------ + 2*y * / 1 - ------------------ | / / 2\ | / / 2\ / / 2\ | \ 4 / / / 2\ | / / 2\ / / 2\ | \ 4 / | | / | y | / | y | | / | y | / | y | | / | y | / | y | / | y | / | y | | / / 2\ / / 2\ | | / | y | / | y | | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | / | y | | / | y | / | y | | / | y | | / | y | / | y | | | | / polar_lift|1 - --| / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| / polar_lift|1 - --| / polar_lift|1 - --| | / | y | / | y | | | / polar_lift|1 - --| / polar_lift|1 - --| | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | | / | y | / | y | | / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | / polar_lift|1 - --| | / polar_lift|1 - --| / polar_lift|1 - --| | | | \/ \ 4 / \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / \/ \ 4 / \/ \ 4 / \/ \ 4 / \/ \ 4 / | / polar_lift|1 - --| / polar_lift|1 - --| | \ \/ \ 4 / \/ \ 4 / / | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | | / polar_lift|1 - --| / polar_lift|1 - --| | \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / \/ \ 4 / \ \/ \ 4 / \/ \ 4 / / | \ \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / \ \/ \ 4 / \/ \ 4 / / | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{24 i x^{3} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} - \frac{72 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}} - \frac{6 i x y^{2} \left(1 - \frac{y^{2}}{4}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{3 \pi x y^{2} \left(1 - \frac{y^{2}}{4}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{6 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} + \frac{24 i x \left(1 - \frac{y^{2}}{4}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{12 \pi x \left(1 - \frac{y^{2}}{4}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{24 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} - \frac{6 i y^{2} \left(1 - \frac{y^{2}}{4}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} + \frac{3 \pi y^{2} \left(1 - \frac{y^{2}}{4}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} - \frac{6 i y^{2} \left(1 - \frac{y^{2}}{4}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} - \frac{6 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}} + \frac{24 i \left(1 - \frac{y^{2}}{4}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} - \frac{12 \pi \left(1 - \frac{y^{2}}{4}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right)^{2}} + \frac{24 i \left(1 - \frac{y^{2}}{4}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} + \frac{24 i \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{\frac{y^{2}}{4} - 1}\right|} > 1 \\\frac{12 x^{3} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right)^{2}} + \frac{36 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}} - \frac{3 x y^{2} \left(1 - \frac{y^{2}}{4}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + \frac{3 x y^{2} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right)^{2}} + \frac{12 x \left(1 - \frac{y^{2}}{4}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{12 x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right)^{2}} + \frac{3 y^{2} \left(1 - \frac{y^{2}}{4}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right)^{2}} + \frac{3 y^{2} \left(1 - \frac{y^{2}}{4}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} + \frac{3 y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}} - \frac{12 \left(1 - \frac{y^{2}}{4}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right)^{2}} - \frac{12 \left(1 - \frac{y^{2}}{4}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}} - \frac{12 \sqrt{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - \frac{y^{2}}{4} \right)}} + 1}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((24*i*sqrt(polar_lift(1 - y^2/4))/(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4))) - 72*i*x^2*sqrt(polar_lift(1 - y^2/4))/(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4))) - 6*i*y^2*sqrt(polar_lift(1 - y^2/4))/(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4))) + 24*i*(1 - y^2/4)/((-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))*sqrt(polar_lift(1 - y^2/4))) - 24*i*x^3*sqrt(polar_lift(1 - y^2/4))*(16*x/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)))/(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))^2 - 12*pi*sqrt(-1 + x^2/polar_lift(1 - y^2/4))*(1 - y^2/4)*(16*x/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)))/(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))^2 - 6*i*y^2*(1 - y^2/4)/((-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))*sqrt(polar_lift(1 - y^2/4))) + 24*i*x*sqrt(polar_lift(1 - y^2/4))*(16*x/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)))/(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))^2 - 12*pi*x*(1 - y^2/4)/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))*polar_lift(1 - y^2/4)) - 6*i*x*y^2*sqrt(polar_lift(1 - y^2/4))*(16*x/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)))/(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))^2 + 3*pi*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4))*(1 - y^2/4)*(16*x/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)))/(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))^2 + 24*i*sqrt(-1 + x^2/polar_lift(1 - y^2/4))*(1 - y^2/4)*(16*x/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)))*acosh(x/sqrt(polar_lift(1 - y^2/4)))/(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))^2 - 6*i*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4))*(1 - y^2/4)*(16*x/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)))*acosh(x/sqrt(polar_lift(1 - y^2/4)))/(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))^2 + 3*pi*x*y^2*(1 - y^2/4)/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))*polar_lift(1 - y^2/4)) + 24*i*x*(1 - y^2/4)*acosh(x/sqrt(polar_lift(1 - y^2/4)))/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))*polar_lift(1 - y^2/4)) - 6*i*x*y^2*(1 - y^2/4)*acosh(x/sqrt(polar_lift(1 - y^2/4)))/(sqrt(-1 + x^2/polar_lift(1 - y^2/4))*(-16*sqrt(-1 + x^2/polar_lift(1 - y^2/4)) + 4*y^2*sqrt(-1 + x^2/polar_lift(1 - y^2/4)))*polar_lift(1 - y^2/4)), x^2/|-1 + y^2/4| > 1), (-12*sqrt(polar_lift(1 - y^2/4))/(-8*sqrt(1 - x^2/polar_lift(1 - y^2/4)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4))) - 12*(1 - y^2/4)/((-8*sqrt(1 - x^2/polar_lift(1 - y^2/4)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4)))*sqrt(polar_lift(1 - y^2/4))) + 3*y^2*sqrt(polar_lift(1 - y^2/4))/(-8*sqrt(1 - x^2/polar_lift(1 - y^2/4)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4))) + 36*x^2*sqrt(polar_lift(1 - y^2/4))/(-8*sqrt(1 - x^2/polar_lift(1 - y^2/4)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4))) - 12*x*sqrt(polar_lift(1 - y^2/4))*(-8*x/(sqrt(1 - x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)))/(-8*sqrt(1 - x^2/polar_lift(1 - y^2/4)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4)))^2 + 3*y^2*(1 - y^2/4)/((-8*sqrt(1 - x^2/polar_lift(1 - y^2/4)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4)))*sqrt(polar_lift(1 - y^2/4))) + 12*x^3*sqrt(polar_lift(1 - y^2/4))*(-8*x/(sqrt(1 - x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)))/(-8*sqrt(1 - x^2/polar_lift(1 - y^2/4)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4)))^2 - 12*sqrt(1 - x^2/polar_lift(1 - y^2/4))*(1 - y^2/4)*(-8*x/(sqrt(1 - x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)))*asin(x/sqrt(polar_lift(1 - y^2/4)))/(-8*sqrt(1 - x^2/polar_lift(1 - y^2/4)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4)))^2 + 3*x*y^2*sqrt(polar_lift(1 - y^2/4))*(-8*x/(sqrt(1 - x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)))/(-8*sqrt(1 - x^2/polar_lift(1 - y^2/4)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4)))^2 + 3*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4))*(1 - y^2/4)*(-8*x/(sqrt(1 - x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(1 - y^2/4))*polar_lift(1 - y^2/4)))*asin(x/sqrt(polar_lift(1 - y^2/4)))/(-8*sqrt(1 - x^2/polar_lift(1 - y^2/4)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4)))^2 + 12*x*(1 - y^2/4)*asin(x/sqrt(polar_lift(1 - y^2/4)))/(sqrt(1 - x^2/polar_lift(1 - y^2/4))*(-8*sqrt(1 - x^2/polar_lift(1 - y^2/4)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4)))*polar_lift(1 - y^2/4)) - 3*x*y^2*(1 - y^2/4)*asin(x/sqrt(polar_lift(1 - y^2/4)))/(sqrt(1 - x^2/polar_lift(1 - y^2/4))*(-8*sqrt(1 - x^2/polar_lift(1 - y^2/4)) + 2*y^2*sqrt(1 - x^2/polar_lift(1 - y^2/4)))*polar_lift(1 - y^2/4)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.