Sr Examen
Integral de (-x^2-y^2+2y+3)^(1/2) dx
Solución
Ha introducido
[src]
1 / | | _____________________ | / 2 2 | \/ - x - y + 2*y + 3 dx | / 0
$$\int\limits_{0}^{1} \sqrt{\left(2 y + \left(- x^{2} - y^{2}\right)\right) + 3}\, dx$$
Integral(sqrt(-x^2 - y^2 + 2*y + 3), (x, 0, 1))
Respuesta (Indefinida)
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// _______________________________ _______________________________ _______________________________ \
|| _______________________________ _______________________________ _______________________________ / 2 / 2 / 2 |
|| / 2 / 2 / 2 / x / 2 \ / x \ 2 / x / 2 \ / x \ / x / 2 \ / x \ |
|| / x / 2 \ 2 / x / 2 \ / x / 2 \ 6*I* / -1 + ------------------------ *\3 - y + 2*y/*acosh|-----------------------------| 2*I*y * / -1 + ------------------------ *\3 - y + 2*y/*acosh|-----------------------------| 4*I*y* / -1 + ------------------------ *\3 - y + 2*y/*acosh|-----------------------------| |
|| 3*pi* / -1 + ------------------------ *\3 - y + 2*y/ __________________________ __________________________ pi*y * / -1 + ------------------------ *\3 - y + 2*y/ 2*pi*y* / -1 + ------------------------ *\3 - y + 2*y/ __________________________ __________________________ / / 2 \ | __________________________| / / 2 \ | __________________________| / / 2 \ | __________________________| |
|| / / 2 \ 3 / / 2 \ / / 2 \ / / 2 \ / / 2 \ 2 / / 2 \ / / 2 \ \/ polar_lift\3 - y + 2*y/ | / / 2 \ | \/ polar_lift\3 - y + 2*y/ | / / 2 \ | \/ polar_lift\3 - y + 2*y/ | / / 2 \ | | 2 | |
|| \/ polar_lift\3 - y + 2*y/ 2*I*x *\/ polar_lift\3 - y + 2*y/ 6*I*x*\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ 2*I*x*y *\/ polar_lift\3 - y + 2*y/ 4*I*x*y*\/ polar_lift\3 - y + 2*y/ \\/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / | x | |
||- ----------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- for |------------| > 1|
|| _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ | 2 | |
|| / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 |3 - y + 2*y| |
/ || / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x |
| || - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |
| _____________________ || / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ |
| / 2 2 || \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ |
| \/ - x - y + 2*y + 3 dx = C + |< |
| || ______________________________ ______________________________ ______________________________ |
/ || / 2 / 2 / 2 |
|| / x / 2 \ / x \ 2 / x / 2 \ / x \ / x / 2 \ / x \ |
|| 3* / 1 - ------------------------ *\3 - y + 2*y/*asin|-----------------------------| y * / 1 - ------------------------ *\3 - y + 2*y/*asin|-----------------------------| 2*y* / 1 - ------------------------ *\3 - y + 2*y/*asin|-----------------------------| |
|| __________________________ __________________________ __________________________ / / 2 \ | __________________________| __________________________ / / 2 \ | __________________________| / / 2 \ | __________________________| |
|| 3 / / 2 \ / / 2 \ 2 / / 2 \ \/ polar_lift\3 - y + 2*y/ | / / 2 \ | / / 2 \ \/ polar_lift\3 - y + 2*y/ | / / 2 \ | \/ polar_lift\3 - y + 2*y/ | / / 2 \ | |
|| x *\/ polar_lift\3 - y + 2*y/ 3*x*\/ polar_lift\3 - y + 2*y/ x*y *\/ polar_lift\3 - y + 2*y/ \\/ polar_lift\3 - y + 2*y/ / 2*x*y*\/ polar_lift\3 - y + 2*y/ \\/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / |
|| ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- otherwise |
|| ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ |
|| / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 |
|| / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x |
|| - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |
|| / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ |
\\ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ /
$$\int \sqrt{\left(2 y + \left(- x^{2} - y^{2}\right)\right) + 3}\, dx = C + \begin{cases} - \frac{2 i x^{3} \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} - \frac{2 i x y^{2} \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} + \frac{4 i x y \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} + \frac{6 i x \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} - \frac{2 i y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} + \frac{\pi y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right)}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} + \frac{4 i y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} - \frac{2 \pi y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right)}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} + \frac{6 i \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} - \frac{3 \pi \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right)}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{- y^{2} + 2 y + 3}}\right| > 1 \\\frac{x^{3} \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} + \frac{x y^{2} \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} - \frac{2 x y \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} - \frac{3 x \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} + \frac{y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(- y^{2} + 2 y + 3\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} - \frac{2 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(- y^{2} + 2 y + 3\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} - \frac{3 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(- y^{2} + 2 y + 3\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} & \text{otherwise} \end{cases}$$
Respuesta
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1 / | | / _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ | | / 2 / 2 \ __________________________ / 2 \ __________________________ / 2 \ / 2 / 2 \ / 2 / 2 \ __________________________ / 2 \ __________________________ / 2 \ / 2 / 2 \ / 2 / 2 \ / 2 / 2 \ | | / x / 2 \ | 12*x 4*x*y 8*x*y | 3 / / 2 \ | 12*x 4*x*y 8*x*y | / / 2 \ | 12*x 4*x*y 8*x*y | 2 / x / 2 \ | 12*x 4*x*y 8*x*y | / x / 2 \ | 12*x 4*x*y 8*x*y | 2 / / 2 \ | 12*x 4*x*y 8*x*y | / / 2 \ | 12*x 4*x*y 8*x*y | / x / 2 \ | 12*x 4*x*y 8*x*y | / x \ 2 / x / 2 \ | 12*x 4*x*y 8*x*y | / x \ / x / 2 \ | 12*x 4*x*y 8*x*y | / x \ | | 3*pi* / -1 + ------------------------ *\3 - y + 2*y/*|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| 2*I*x *\/ polar_lift\3 - y + 2*y/ *|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| 6*I*x*\/ polar_lift\3 - y + 2*y/ *|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| pi*y * / -1 + ------------------------ *\3 - y + 2*y/*|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| 2*pi*y* / -1 + ------------------------ *\3 - y + 2*y/*|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| 2*I*x*y *\/ polar_lift\3 - y + 2*y/ *|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| 4*I*x*y*\/ polar_lift\3 - y + 2*y/ *|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| 6*I* / -1 + ------------------------ *\3 - y + 2*y/*|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------|*acosh|-----------------------------| 2*I*y * / -1 + ------------------------ *\3 - y + 2*y/*|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------|*acosh|-----------------------------| 4*I*y* / -1 + ------------------------ *\3 - y + 2*y/*|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------|*acosh|-----------------------------| | | / / 2 \ | _______________________________ _______________________________ _______________________________ | | _______________________________ _______________________________ _______________________________ | | _______________________________ _______________________________ _______________________________ | / / 2 \ | _______________________________ _______________________________ _______________________________ | / / 2 \ | _______________________________ _______________________________ _______________________________ | | _______________________________ _______________________________ _______________________________ | | _______________________________ _______________________________ _______________________________ | / / 2 \ | _______________________________ _______________________________ _______________________________ | | __________________________| / / 2 \ | _______________________________ _______________________________ _______________________________ | | __________________________| / / 2 \ | _______________________________ _______________________________ _______________________________ | | __________________________| | | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / / 2 \ | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / / 2 \ | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / / 2 \ | / 2 \ / x \ 2 / 2 \ / x \ / 2 \ / x \ | | | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| \\/ polar_lift\3 - y + 2*y/ / | / x / 2 \ / x / 2 \ / x / 2 \| \\/ polar_lift\3 - y + 2*y/ / | / x / 2 \ / x / 2 \ / x / 2 \| \\/ polar_lift\3 - y + 2*y/ / 6*I*x*\3 - y + 2*y/*acosh|-----------------------------| 2*I*x*y *\3 - y + 2*y/*acosh|-----------------------------| 4*I*x*y*\3 - y + 2*y/*acosh|-----------------------------| | | __________________________ __________________________ __________________________ __________________________ | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | __________________________| | __________________________| | __________________________| | | / / 2 \ 2 / / 2 \ 2 / / 2 \ / / 2 \ / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | 2 / 2 \ / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | 2 / 2 \ / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ | | / / 2 \ | | / / 2 \ | 2 | | 6*I*\/ polar_lift\3 - y + 2*y/ 6*I*x *\/ polar_lift\3 - y + 2*y/ 2*I*y *\/ polar_lift\3 - y + 2*y/ 4*I*y*\/ polar_lift\3 - y + 2*y/ 6*I*\3 - y + 2*y/ \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / 2*I*y *\3 - y + 2*y/ 4*I*y*\3 - y + 2*y/ \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / 3*pi*x*\3 - y + 2*y/ \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / pi*x*y *\3 - y + 2*y/ 2*pi*x*y*\3 - y + 2*y/ \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / x | |----------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - 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---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- for -------------- > 1 | | _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ / _______________________________ _______________________________ _______________________________\ 2 2 / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ 2 2 _______________________________ / _______________________________ _______________________________ _______________________________\ 2 2 2 2 _______________________________ / _______________________________ _______________________________ _______________________________\ _______________________________ / _______________________________ _______________________________ _______________________________\ 2 2 _______________________________ / _______________________________ _______________________________ _______________________________\ _______________________________ / _______________________________ _______________________________ _______________________________\ _______________________________ / _______________________________ _______________________________ _______________________________\ | 2 | | | / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 | / 2 / 2 / 2 | __________________________ / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ | / 2 / 2 / 2 | __________________________ | / 2 / 2 / 2 | __________________________ / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ / 2 | / 2 / 2 / 2 | / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ / 2 | / 2 / 2 / 2 | / 2 | / 2 / 2 / 2 | / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ / 2 | / 2 / 2 / 2 | / 2 | / 2 / 2 / 2 | / 2 | / 2 / 2 / 2 | |3 - y + 2*y| | | / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x | / x / x 2 / x | / / 2 \ | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / x / x 2 / x | / / 2 \ | / x / x 2 / x | / / 2 \ | / 2 / 2 / 2 | | / 2 / 2 / 2 | / x | / x / x 2 / x | / 2 \ | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | / x | / x / x 2 / x | / 2 \ / x | / x / x 2 / x | / 2 \ | / 2 / 2 / 2 | | / 2 / 2 / 2 | / x | / x / x 2 / x | / 2 \ / x | / x / x 2 / x | / 2 \ / x | / x / x 2 / x | / 2 \ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*\/ polar_lift\3 - y + 2*y/ | / x / x 2 / x | | / x / x 2 / x | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*\/ polar_lift\3 - y + 2*y/ |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*\/ polar_lift\3 - y + 2*y/ | / x / x 2 / x | | / x / x 2 / x | / -1 + ------------------------ *|- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*polar_lift\3 - y + 2*y/ | / x / x 2 / x | | / x / x 2 / x | | / x / x 2 / x | | / x / x 2 / x | / -1 + ------------------------ *|- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*polar_lift\3 - y + 2*y/ / -1 + ------------------------ *|- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*polar_lift\3 - y + 2*y/ | / x / x 2 / x | | / x / x 2 / x | / -1 + ------------------------ *|- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*polar_lift\3 - y + 2*y/ / -1 + ------------------------ *|- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*polar_lift\3 - y + 2*y/ / -1 + ------------------------ *|- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*polar_lift\3 - y + 2*y/ | | / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | | \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | | \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | < dx | | ______________________________ ______________________________ ______________________________ | | __________________________ / 2 \ __________________________ / 2 \ __________________________ / 2 \ / 2 / 2 \ __________________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | 3 / / 2 \ | 6*x 4*x*y 2*x*y | / / 2 \ | 6*x 4*x*y 2*x*y | 2 / / 2 \ | 6*x 4*x*y 2*x*y | / x / 2 \ | 6*x 4*x*y 2*x*y | / x \ / / 2 \ | 6*x 4*x*y 2*x*y | 2 / x / 2 \ | 6*x 4*x*y 2*x*y | / x \ / x / 2 \ | 6*x 4*x*y 2*x*y | / x \ | | x *\/ polar_lift\3 - y + 2*y/ *|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------| 3*x*\/ polar_lift\3 - y + 2*y/ *|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------| x*y *\/ polar_lift\3 - y + 2*y/ *|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------| 3* / 1 - ------------------------ *\3 - y + 2*y/*|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------|*asin|-----------------------------| 2*x*y*\/ polar_lift\3 - y + 2*y/ *|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------| y * / 1 - ------------------------ *\3 - y + 2*y/*|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------|*asin|-----------------------------| 2*y* / 1 - ------------------------ *\3 - y + 2*y/*|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------|*asin|-----------------------------| | | | ______________________________ ______________________________ ______________________________ | | ______________________________ ______________________________ ______________________________ | | ______________________________ ______________________________ ______________________________ | / / 2 \ | ______________________________ ______________________________ ______________________________ | | __________________________| | ______________________________ ______________________________ ______________________________ | / / 2 \ | ______________________________ ______________________________ ______________________________ | | __________________________| / / 2 \ | ______________________________ ______________________________ ______________________________ | | __________________________| | | | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / / 2 \ | | / 2 / 2 / 2 | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / / 2 \ | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / / 2 \ | / 2 \ / x \ 2 / 2 \ / x \ / 2 \ / x \ | | | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| \\/ polar_lift\3 - y + 2*y/ / | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| \\/ polar_lift\3 - y + 2*y/ / | / x / 2 \ / x / 2 \ / x / 2 \| \\/ polar_lift\3 - y + 2*y/ / 3*x*\3 - y + 2*y/*asin|-----------------------------| x*y *\3 - y + 2*y/*asin|-----------------------------| 2*x*y*\3 - y + 2*y/*asin|-----------------------------| | | __________________________ __________________________ __________________________ __________________________ | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | __________________________| | __________________________| | __________________________| | | / / 2 \ 2 / / 2 \ / 2 \ / / 2 \ 2 / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | 2 / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ | | / / 2 \ | | / / 2 \ | | | 3*\/ polar_lift\3 - y + 2*y/ y *\/ polar_lift\3 - y + 2*y/ 3*\3 - y + 2*y/ 2*y*\/ polar_lift\3 - y + 2*y/ 3*x *\/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / y *\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / 2*y*\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / | | - ------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - 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--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- otherwise | | ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ / ______________________________ ______________________________ ______________________________\ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ 2 / ______________________________ ______________________________ ______________________________\ 2 / ______________________________ ______________________________ ______________________________\ 2 2 2 2 2 ______________________________ / ______________________________ ______________________________ ______________________________\ ______________________________ / ______________________________ ______________________________ ______________________________\ ______________________________ / ______________________________ ______________________________ ______________________________\ | | / 2 / 2 / 2 / 2 / 2 / 2 | / 2 / 2 / 2 | __________________________ / 2 / 2 / 2 / 2 / 2 / 2 / ______________________________ ______________________________ ______________________________\ | / 2 / 2 / 2 | __________________________ / ______________________________ ______________________________ ______________________________\ | / 2 / 2 / 2 | __________________________ / ______________________________ ______________________________ ______________________________\ / ______________________________ ______________________________ ______________________________\ / ______________________________ ______________________________ ______________________________\ / ______________________________ ______________________________ ______________________________\ / ______________________________ ______________________________ ______________________________\ / 2 | / 2 / 2 / 2 | / 2 | / 2 / 2 / 2 | / 2 | / 2 / 2 / 2 | | | / x / x 2 / x / x / x 2 / x | / x / x 2 / x | / / 2 \ / x / x 2 / x / x / x 2 / x | / 2 / 2 / 2 | | / x / x 2 / x | / / 2 \ | / 2 / 2 / 2 | | / x / x 2 / x | / / 2 \ | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | / x | / x / x 2 / x | / 2 \ / x | / x / x 2 / x | / 2 \ / x | / x / x 2 / x | / 2 \ | | - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |*\/ polar_lift\3 - y + 2*y/ - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | / x / x 2 / x | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |*\/ polar_lift\3 - y + 2*y/ | / x / x 2 / x | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |*\/ polar_lift\3 - y + 2*y/ | / x / x 2 / x | | / x / x 2 / x | | / x / x 2 / x | | / x / x 2 / x | | / x / x 2 / x | / 1 - ------------------------ *|- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |*polar_lift\3 - y + 2*y/ / 1 - ------------------------ *|- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |*polar_lift\3 - y + 2*y/ / 1 - ------------------------ *|- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |*polar_lift\3 - y + 2*y/ | | / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | | / / 2 \ / / 2 \ / / 2 \ | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | | / / 2 \ / / 2 \ / / 2 \ | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | | \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ | / / 2 \ / / 2 \ / / 2 \ | \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / / 2 \ / / 2 \ / / 2 \ | \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | \ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{2 i x^{3} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} - \frac{2 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} - \frac{2 i x y^{2} \left(- y^{2} + 2 y + 3\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{\pi x y^{2} \left(- y^{2} + 2 y + 3\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{4 i x y \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} + \frac{4 i x y \left(- y^{2} + 2 y + 3\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{2 \pi x y \left(- y^{2} + 2 y + 3\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{6 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} + \frac{6 i x \left(- y^{2} + 2 y + 3\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{3 \pi x \left(- y^{2} + 2 y + 3\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{2 i y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} + \frac{\pi y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(- y^{2} + 2 y + 3\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} - \frac{2 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} + \frac{4 i y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} - \frac{2 \pi y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} + \frac{4 i y \left(- y^{2} + 2 y + 3\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} + \frac{4 i y \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} + \frac{6 i \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} - \frac{3 \pi \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} + \frac{6 i \left(- y^{2} + 2 y + 3\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} + \frac{6 i \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{- y^{2} + 2 y + 3}\right|} > 1 \\\frac{x^{3} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} + \frac{3 x^{2} \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} + \frac{x y^{2} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} - \frac{x y^{2} \left(- y^{2} + 2 y + 3\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{2 x y \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} + \frac{2 x y \left(- y^{2} + 2 y + 3\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{3 x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} + \frac{3 x \left(- y^{2} + 2 y + 3\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(- y^{2} + 2 y + 3\right) \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(- y^{2} + 2 y + 3\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} + \frac{y^{2} \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} - \frac{2 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(- y^{2} + 2 y + 3\right) \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} - \frac{2 y \left(- y^{2} + 2 y + 3\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} - \frac{2 y \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} - \frac{3 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(- y^{2} + 2 y + 3\right) \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} - \frac{3 \left(- y^{2} + 2 y + 3\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} - \frac{3 \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} & \text{otherwise} \end{cases}\, dx$$
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1 / | | / _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ | | / 2 / 2 \ __________________________ / 2 \ __________________________ / 2 \ / 2 / 2 \ / 2 / 2 \ __________________________ / 2 \ __________________________ / 2 \ / 2 / 2 \ / 2 / 2 \ / 2 / 2 \ | | / x / 2 \ | 12*x 4*x*y 8*x*y | 3 / / 2 \ | 12*x 4*x*y 8*x*y | / / 2 \ | 12*x 4*x*y 8*x*y | 2 / x / 2 \ | 12*x 4*x*y 8*x*y | / x / 2 \ | 12*x 4*x*y 8*x*y | 2 / / 2 \ | 12*x 4*x*y 8*x*y | / / 2 \ | 12*x 4*x*y 8*x*y | / x / 2 \ | 12*x 4*x*y 8*x*y | / x \ 2 / x / 2 \ | 12*x 4*x*y 8*x*y | / x \ / x / 2 \ | 12*x 4*x*y 8*x*y | / x \ | | 3*pi* / -1 + ------------------------ *\3 - y + 2*y/*|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| 2*I*x *\/ polar_lift\3 - y + 2*y/ *|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| 6*I*x*\/ polar_lift\3 - y + 2*y/ *|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| pi*y * / -1 + ------------------------ *\3 - y + 2*y/*|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| 2*pi*y* / -1 + ------------------------ *\3 - y + 2*y/*|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| 2*I*x*y *\/ polar_lift\3 - y + 2*y/ *|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| 4*I*x*y*\/ polar_lift\3 - y + 2*y/ *|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------| 6*I* / -1 + ------------------------ *\3 - y + 2*y/*|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------|*acosh|-----------------------------| 2*I*y * / -1 + ------------------------ *\3 - y + 2*y/*|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------|*acosh|-----------------------------| 4*I*y* / -1 + ------------------------ *\3 - y + 2*y/*|-------------------------------------------------------------- - -------------------------------------------------------------- + --------------------------------------------------------------|*acosh|-----------------------------| | | / / 2 \ | _______________________________ _______________________________ _______________________________ | | _______________________________ _______________________________ _______________________________ | | _______________________________ _______________________________ _______________________________ | / / 2 \ | _______________________________ _______________________________ _______________________________ | / / 2 \ | _______________________________ _______________________________ _______________________________ | | _______________________________ _______________________________ _______________________________ | | _______________________________ _______________________________ _______________________________ | / / 2 \ | _______________________________ _______________________________ _______________________________ | | __________________________| / / 2 \ | _______________________________ _______________________________ _______________________________ | | __________________________| / / 2 \ | _______________________________ _______________________________ _______________________________ | | __________________________| | | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / / 2 \ | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / / 2 \ | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / / 2 \ | / 2 \ / x \ 2 / 2 \ / x \ / 2 \ / x \ | | | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| \\/ polar_lift\3 - y + 2*y/ / | / x / 2 \ / x / 2 \ / x / 2 \| \\/ polar_lift\3 - y + 2*y/ / | / x / 2 \ / x / 2 \ / x / 2 \| \\/ polar_lift\3 - y + 2*y/ / 6*I*x*\3 - y + 2*y/*acosh|-----------------------------| 2*I*x*y *\3 - y + 2*y/*acosh|-----------------------------| 4*I*x*y*\3 - y + 2*y/*acosh|-----------------------------| | | __________________________ __________________________ __________________________ __________________________ | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/ / -1 + ------------------------ *polar_lift\3 - y + 2*y/| | __________________________| | __________________________| | __________________________| | | / / 2 \ 2 / / 2 \ 2 / / 2 \ / / 2 \ / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | 2 / 2 \ / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | 2 / 2 \ / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ | | / / 2 \ | | / / 2 \ | 2 | | 6*I*\/ polar_lift\3 - y + 2*y/ 6*I*x *\/ polar_lift\3 - y + 2*y/ 2*I*y *\/ polar_lift\3 - y + 2*y/ 4*I*y*\/ polar_lift\3 - y + 2*y/ 6*I*\3 - y + 2*y/ \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / 2*I*y *\3 - y + 2*y/ 4*I*y*\3 - y + 2*y/ \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / 3*pi*x*\3 - y + 2*y/ \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / pi*x*y *\3 - y + 2*y/ 2*pi*x*y*\3 - y + 2*y/ \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / x | |----------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - 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---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- for -------------- > 1 | | _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ / _______________________________ _______________________________ _______________________________\ 2 2 / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ 2 2 _______________________________ / _______________________________ _______________________________ _______________________________\ 2 2 2 2 _______________________________ / _______________________________ _______________________________ _______________________________\ _______________________________ / _______________________________ _______________________________ _______________________________\ 2 2 _______________________________ / _______________________________ _______________________________ _______________________________\ _______________________________ / _______________________________ _______________________________ _______________________________\ _______________________________ / _______________________________ _______________________________ _______________________________\ | 2 | | | / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 | / 2 / 2 / 2 | __________________________ / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ | / 2 / 2 / 2 | __________________________ | / 2 / 2 / 2 | __________________________ / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ / 2 | / 2 / 2 / 2 | / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ / 2 | / 2 / 2 / 2 | / 2 | / 2 / 2 / 2 | / _______________________________ _______________________________ _______________________________\ / _______________________________ _______________________________ _______________________________\ / 2 | / 2 / 2 / 2 | / 2 | / 2 / 2 / 2 | / 2 | / 2 / 2 / 2 | |3 - y + 2*y| | | / x / x 2 / x / x / x 2 / x / x / x 2 / x / x / x 2 / x | / x / x 2 / x | / / 2 \ | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / x / x 2 / x | / / 2 \ | / x / x 2 / x | / / 2 \ | / 2 / 2 / 2 | | / 2 / 2 / 2 | / x | / x / x 2 / x | / 2 \ | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | / x | / x / x 2 / x | / 2 \ / x | / x / x 2 / x | / 2 \ | / 2 / 2 / 2 | | / 2 / 2 / 2 | / x | / x / x 2 / x | / 2 \ / x | / x / x 2 / x | / 2 \ / x | / x / x 2 / x | / 2 \ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ - 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*\/ polar_lift\3 - y + 2*y/ | / x / x 2 / x | | / x / x 2 / x | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*\/ polar_lift\3 - y + 2*y/ |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*\/ polar_lift\3 - y + 2*y/ | / x / x 2 / x | | / x / x 2 / x | / -1 + ------------------------ *|- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*polar_lift\3 - y + 2*y/ | / x / x 2 / x | | / x / x 2 / x | | / x / x 2 / x | | / x / x 2 / x | / -1 + ------------------------ *|- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*polar_lift\3 - y + 2*y/ / -1 + ------------------------ *|- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*polar_lift\3 - y + 2*y/ | / x / x 2 / x | | / x / x 2 / x | / -1 + ------------------------ *|- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*polar_lift\3 - y + 2*y/ / -1 + ------------------------ *|- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*polar_lift\3 - y + 2*y/ / -1 + ------------------------ *|- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ |*polar_lift\3 - y + 2*y/ | | / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | |- 12* / -1 + ------------------------ - 8*y* / -1 + ------------------------ + 4*y * / -1 + ------------------------ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | | \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | | \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | < dx | | ______________________________ ______________________________ ______________________________ | | __________________________ / 2 \ __________________________ / 2 \ __________________________ / 2 \ / 2 / 2 \ __________________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | 3 / / 2 \ | 6*x 4*x*y 2*x*y | / / 2 \ | 6*x 4*x*y 2*x*y | 2 / / 2 \ | 6*x 4*x*y 2*x*y | / x / 2 \ | 6*x 4*x*y 2*x*y | / x \ / / 2 \ | 6*x 4*x*y 2*x*y | 2 / x / 2 \ | 6*x 4*x*y 2*x*y | / x \ / x / 2 \ | 6*x 4*x*y 2*x*y | / x \ | | x *\/ polar_lift\3 - y + 2*y/ *|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------| 3*x*\/ polar_lift\3 - y + 2*y/ *|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------| x*y *\/ polar_lift\3 - y + 2*y/ *|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------| 3* / 1 - ------------------------ *\3 - y + 2*y/*|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------|*asin|-----------------------------| 2*x*y*\/ polar_lift\3 - y + 2*y/ *|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------| y * / 1 - ------------------------ *\3 - y + 2*y/*|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------|*asin|-----------------------------| 2*y* / 1 - ------------------------ *\3 - y + 2*y/*|- ------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------|*asin|-----------------------------| | | | ______________________________ ______________________________ ______________________________ | | ______________________________ ______________________________ ______________________________ | | ______________________________ ______________________________ ______________________________ | / / 2 \ | ______________________________ ______________________________ ______________________________ | | __________________________| | ______________________________ ______________________________ ______________________________ | / / 2 \ | ______________________________ ______________________________ ______________________________ | | __________________________| / / 2 \ | ______________________________ ______________________________ ______________________________ | | __________________________| | | | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / / 2 \ | | / 2 / 2 / 2 | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / / 2 \ | \/ polar_lift\3 - y + 2*y/ | / 2 / 2 / 2 | | / / 2 \ | / 2 \ / x \ 2 / 2 \ / x \ / 2 \ / x \ | | | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| \\/ polar_lift\3 - y + 2*y/ / | / x / 2 \ / x / 2 \ / x / 2 \| | / x / 2 \ / x / 2 \ / x / 2 \| \\/ polar_lift\3 - y + 2*y/ / | / x / 2 \ / x / 2 \ / x / 2 \| \\/ polar_lift\3 - y + 2*y/ / 3*x*\3 - y + 2*y/*asin|-----------------------------| x*y *\3 - y + 2*y/*asin|-----------------------------| 2*x*y*\3 - y + 2*y/*asin|-----------------------------| | | __________________________ __________________________ __________________________ __________________________ | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/ / 1 - ------------------------ *polar_lift\3 - y + 2*y/| | __________________________| | __________________________| | __________________________| | | / / 2 \ 2 / / 2 \ / 2 \ / / 2 \ 2 / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | 2 / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ | | / / 2 \ | | / / 2 \ | | | 3*\/ polar_lift\3 - y + 2*y/ y *\/ polar_lift\3 - y + 2*y/ 3*\3 - y + 2*y/ 2*y*\/ polar_lift\3 - y + 2*y/ 3*x *\/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / y *\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / 2*y*\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / \\/ polar_lift\3 - y + 2*y/ / | | - ------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- otherwise | | ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ / ______________________________ ______________________________ ______________________________\ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ 2 / ______________________________ ______________________________ ______________________________\ 2 / ______________________________ ______________________________ ______________________________\ 2 2 2 2 2 ______________________________ / ______________________________ ______________________________ ______________________________\ ______________________________ / ______________________________ ______________________________ ______________________________\ ______________________________ / ______________________________ ______________________________ ______________________________\ | | / 2 / 2 / 2 / 2 / 2 / 2 | / 2 / 2 / 2 | __________________________ / 2 / 2 / 2 / 2 / 2 / 2 / ______________________________ ______________________________ ______________________________\ | / 2 / 2 / 2 | __________________________ / ______________________________ ______________________________ ______________________________\ | / 2 / 2 / 2 | __________________________ / ______________________________ ______________________________ ______________________________\ / ______________________________ ______________________________ ______________________________\ / ______________________________ ______________________________ ______________________________\ / ______________________________ ______________________________ ______________________________\ / ______________________________ ______________________________ ______________________________\ / 2 | / 2 / 2 / 2 | / 2 | / 2 / 2 / 2 | / 2 | / 2 / 2 / 2 | | | / x / x 2 / x / x / x 2 / x | / x / x 2 / x | / / 2 \ / x / x 2 / x / x / x 2 / x | / 2 / 2 / 2 | | / x / x 2 / x | / / 2 \ | / 2 / 2 / 2 | | / x / x 2 / x | / / 2 \ | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | | / 2 / 2 / 2 | / x | / x / x 2 / x | / 2 \ / x | / x / x 2 / x | / 2 \ / x | / x / x 2 / x | / 2 \ | | - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |*\/ polar_lift\3 - y + 2*y/ - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ - 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | / x / x 2 / x | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |*\/ polar_lift\3 - y + 2*y/ | / x / x 2 / x | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |*\/ polar_lift\3 - y + 2*y/ | / x / x 2 / x | | / x / x 2 / x | | / x / x 2 / x | | / x / x 2 / x | | / x / x 2 / x | / 1 - ------------------------ *|- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |*polar_lift\3 - y + 2*y/ / 1 - ------------------------ *|- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |*polar_lift\3 - y + 2*y/ / 1 - ------------------------ *|- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ |*polar_lift\3 - y + 2*y/ | | / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ / / 2 \ |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | | / / 2 \ / / 2 \ / / 2 \ | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | | / / 2 \ / / 2 \ / / 2 \ | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | |- 6* / 1 - ------------------------ - 4*y* / 1 - ------------------------ + 2*y * / 1 - ------------------------ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | / / 2 \ | / / 2 \ / / 2 \ / / 2 \ | | | \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ | / / 2 \ / / 2 \ / / 2 \ | \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / / 2 \ / / 2 \ / / 2 \ | \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | | / / 2 \ / / 2 \ / / 2 \ | \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \/ polar_lift\3 - y + 2*y/ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | \ \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / \ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ \/ polar_lift\3 - y + 2*y/ / | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{2 i x^{3} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} - \frac{2 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} - \frac{2 i x y^{2} \left(- y^{2} + 2 y + 3\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{\pi x y^{2} \left(- y^{2} + 2 y + 3\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{4 i x y \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} + \frac{4 i x y \left(- y^{2} + 2 y + 3\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{2 \pi x y \left(- y^{2} + 2 y + 3\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{6 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} + \frac{6 i x \left(- y^{2} + 2 y + 3\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{3 \pi x \left(- y^{2} + 2 y + 3\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{2 i y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} + \frac{\pi y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(- y^{2} + 2 y + 3\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} - \frac{2 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} + \frac{4 i y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} - \frac{2 \pi y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} + \frac{4 i y \left(- y^{2} + 2 y + 3\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} + \frac{4 i y \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} + \frac{6 i \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} - \frac{3 \pi \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \left(- y^{2} + 2 y + 3\right) \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{8 x y}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{12 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right)^{2}} + \frac{6 i \left(- y^{2} + 2 y + 3\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} + \frac{6 i \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 8 y \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1} - 12 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{- y^{2} + 2 y + 3}\right|} > 1 \\\frac{x^{3} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} + \frac{3 x^{2} \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} + \frac{x y^{2} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} - \frac{x y^{2} \left(- y^{2} + 2 y + 3\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{2 x y \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} + \frac{2 x y \left(- y^{2} + 2 y + 3\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{3 x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} + \frac{3 x \left(- y^{2} + 2 y + 3\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right) \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + \frac{y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(- y^{2} + 2 y + 3\right) \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(- y^{2} + 2 y + 3\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} + \frac{y^{2} \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} - \frac{2 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(- y^{2} + 2 y + 3\right) \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} - \frac{2 y \left(- y^{2} + 2 y + 3\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} - \frac{2 y \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} - \frac{3 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \left(- y^{2} + 2 y + 3\right) \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{4 x y}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} - \frac{6 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} \operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right)^{2}} - \frac{3 \left(- y^{2} + 2 y + 3\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}} - \frac{3 \sqrt{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 4 y \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1} - 6 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(- y^{2} + 2 y + 3 \right)}} + 1}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((6*i*sqrt(polar_lift(3 - y^2 + 2*y))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))) - 6*i*x^2*sqrt(polar_lift(3 - y^2 + 2*y))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))) - 2*i*y^2*sqrt(polar_lift(3 - y^2 + 2*y))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))) + 4*i*y*sqrt(polar_lift(3 - y^2 + 2*y))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))) + 6*i*(3 - y^2 + 2*y)/((-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))*sqrt(polar_lift(3 - y^2 + 2*y))) - 3*pi*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*(3 - y^2 + 2*y)*(12*x/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 8*x*y/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))^2 - 2*i*x^3*sqrt(polar_lift(3 - y^2 + 2*y))*(12*x/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 8*x*y/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))^2 - 2*i*y^2*(3 - y^2 + 2*y)/((-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))*sqrt(polar_lift(3 - y^2 + 2*y))) + 4*i*y*(3 - y^2 + 2*y)/((-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))*sqrt(polar_lift(3 - y^2 + 2*y))) + 6*i*x*sqrt(polar_lift(3 - y^2 + 2*y))*(12*x/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 8*x*y/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))^2 + pi*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*(3 - y^2 + 2*y)*(12*x/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 8*x*y/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))^2 - 3*pi*x*(3 - y^2 + 2*y)/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))*polar_lift(3 - y^2 + 2*y)) - 2*pi*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*(3 - y^2 + 2*y)*(12*x/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 8*x*y/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))^2 - 2*i*x*y^2*sqrt(polar_lift(3 - y^2 + 2*y))*(12*x/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 8*x*y/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))^2 + 4*i*x*y*sqrt(polar_lift(3 - y^2 + 2*y))*(12*x/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 8*x*y/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))^2 + 6*i*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*(3 - y^2 + 2*y)*(12*x/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 8*x*y/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))*acosh(x/sqrt(polar_lift(3 - y^2 + 2*y)))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))^2 + pi*x*y^2*(3 - y^2 + 2*y)/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))*polar_lift(3 - y^2 + 2*y)) - 2*pi*x*y*(3 - y^2 + 2*y)/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))*polar_lift(3 - y^2 + 2*y)) - 2*i*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*(3 - y^2 + 2*y)*(12*x/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 8*x*y/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))*acosh(x/sqrt(polar_lift(3 - y^2 + 2*y)))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))^2 + 4*i*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*(3 - y^2 + 2*y)*(12*x/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 8*x*y/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))*acosh(x/sqrt(polar_lift(3 - y^2 + 2*y)))/(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))^2 + 6*i*x*(3 - y^2 + 2*y)*acosh(x/sqrt(polar_lift(3 - y^2 + 2*y)))/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))*polar_lift(3 - y^2 + 2*y)) - 2*i*x*y^2*(3 - y^2 + 2*y)*acosh(x/sqrt(polar_lift(3 - y^2 + 2*y)))/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))*polar_lift(3 - y^2 + 2*y)) + 4*i*x*y*(3 - y^2 + 2*y)*acosh(x/sqrt(polar_lift(3 - y^2 + 2*y)))/(sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y))*(-12*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) - 8*y*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)) + 4*y^2*sqrt(-1 + x^2/polar_lift(3 - y^2 + 2*y)))*polar_lift(3 - y^2 + 2*y)), x^2/|3 - y^2 + 2*y| > 1), (-3*sqrt(polar_lift(3 - y^2 + 2*y))/(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))) + y^2*sqrt(polar_lift(3 - y^2 + 2*y))/(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))) - 3*(3 - y^2 + 2*y)/((-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))*sqrt(polar_lift(3 - y^2 + 2*y))) - 2*y*sqrt(polar_lift(3 - y^2 + 2*y))/(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))) + 3*x^2*sqrt(polar_lift(3 - y^2 + 2*y))/(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))) + x^3*sqrt(polar_lift(3 - y^2 + 2*y))*(-6*x/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))/(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))^2 + y^2*(3 - y^2 + 2*y)/((-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))*sqrt(polar_lift(3 - y^2 + 2*y))) - 3*x*sqrt(polar_lift(3 - y^2 + 2*y))*(-6*x/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))/(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))^2 - 2*y*(3 - y^2 + 2*y)/((-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))*sqrt(polar_lift(3 - y^2 + 2*y))) + x*y^2*sqrt(polar_lift(3 - y^2 + 2*y))*(-6*x/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))/(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))^2 - 3*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*(3 - y^2 + 2*y)*(-6*x/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))*asin(x/sqrt(polar_lift(3 - y^2 + 2*y)))/(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))^2 - 2*x*y*sqrt(polar_lift(3 - y^2 + 2*y))*(-6*x/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))/(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))^2 + y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*(3 - y^2 + 2*y)*(-6*x/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))*asin(x/sqrt(polar_lift(3 - y^2 + 2*y)))/(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))^2 - 2*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*(3 - y^2 + 2*y)*(-6*x/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) - 4*x*y/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*polar_lift(3 - y^2 + 2*y)))*asin(x/sqrt(polar_lift(3 - y^2 + 2*y)))/(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))^2 + 3*x*(3 - y^2 + 2*y)*asin(x/sqrt(polar_lift(3 - y^2 + 2*y)))/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))*polar_lift(3 - y^2 + 2*y)) - x*y^2*(3 - y^2 + 2*y)*asin(x/sqrt(polar_lift(3 - y^2 + 2*y)))/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))*polar_lift(3 - y^2 + 2*y)) + 2*x*y*(3 - y^2 + 2*y)*asin(x/sqrt(polar_lift(3 - y^2 + 2*y)))/(sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y))*(-6*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) - 4*y*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)) + 2*y^2*sqrt(1 - x^2/polar_lift(3 - y^2 + 2*y)))*polar_lift(3 - y^2 + 2*y)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.