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