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Identical expressions
- xy(one -x-y)^ one / two
- xy(1 minus x minus y) to the power of 1 divide by 2
- xy(one minus x minus y) to the power of one divide by two
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- xy(1-x-y)^1/2dx
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Similar expressions
- xy(1-x+y)^1/2
- xy(1+x-y)^1/2
Integral of xy(1-x-y)^1/2 dx
The solution
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[src]
1 - y / | | ___________ | x*y*\/ 1 - x - y dx | / 0
$$\int\limits_{0}^{1 - y} x y \sqrt{- y + \left(1 - x\right)}\, dx$$
Integral((x*y)*sqrt(1 - x - y), (x, 0, 1 - y))
The answer (Indefinite)
[src]
/ | / 5/2 3/2 3/2\ | ___________ | (1 - x - y) (1 - x - y) y*(1 - x - y) | | x*y*\/ 1 - x - y dx = C - 2*y*|- -------------- + -------------- - ----------------| | \ 5 3 3 / /
$$\int x y \sqrt{- y + \left(1 - x\right)}\, dx = C - 2 y \left(- \frac{y \left(- x - y + 1\right)^{\frac{3}{2}}}{3} - \frac{\left(- x - y + 1\right)^{\frac{5}{2}}}{5} + \frac{\left(- x - y + 1\right)^{\frac{3}{2}}}{3}\right)$$
The answer
[src]
/ 0 | / | | | | / / ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ \ | | | | / x 5/2 / x 3/2 / x 5/2 / x 5/2 3 / x ___________________ 2 / x ___________________ / x 3/2 / x ___________________ / x 3/2 2 / x ___________________ / x 5/2 / x 3/2 2 / x 3/2 / x 3/2 / x ___________________ 2 / x ___________________| | | | | 5/2 5/2 5/2 5/2 60*I* / -1 + ----------------- *polar_lift (1 - y) 8*I* / -1 + ----------------- *polar_lift (1 - y) 3/2 6*I* / -1 + ----------------- *polar_lift (1 - y) 90*I*x* / -1 + ----------------- *polar_lift (1 - y) 90*I*x * / -1 + ----------------- *\/ polar_lift(1 - y) 60*I*x * / -1 + ----------------- *\/ polar_lift(1 - y) 24*I*x* / -1 + ----------------- *polar_lift (1 - y) 2 ___________________ ___________________ 3/2 2 3/2 3 8*I*x* / -1 + ----------------- *\/ polar_lift(1 - y) 8*I*y* / -1 + ----------------- *polar_lift (1 - y) 18*I*x * / -1 + ----------------- *\/ polar_lift(1 - y) 60*I*y* / -1 + ----------------- *polar_lift (1 - y) 120*I*x* / -1 + ----------------- *polar_lift (1 - y) 180*I*x * / -1 + ----------------- *polar_lift (1 - y) 120*I*x*y* / -1 + ----------------- *polar_lift (1 - y) 8*I*x*y* / -1 + ----------------- *\/ polar_lift(1 - y) 2 ___________________ 60*I*y*x * / -1 + ----------------- *\/ polar_lift(1 - y) | | | | |4*polar_lift (1 - y) 60*polar_lift (1 - y) 60*x*polar_lift (1 - y) 60*y*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 2*I*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 6*I*x *\/ polar_lift(1 - y) 4*I*x*\/ polar_lift(1 - y) 2*I*y*polar_lift (1 - y) 2*I*x 3*I*x*polar_lift (1 - y) 3*I*x \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 2*I*y*x 4*I*x*y*\/ polar_lift(1 - y) \/ polar_lift(1 - y) | | x | | | |y*|---------------------- + ----------------------- - ------------------------- - ------------------------- - ------------------------------------------------------ - ----------------------------------------------------- + ------------------------------------------------ + ----------------------------------------------------- - -------------------------------------------------------- - ---------------------------------------------------------- - ---------------------------------------------------------- - -------------------------------------------------------- - ------------------------------------------------ - ------------------------------------------------ - ------------------------------------------------ + ---------------------------------------------------------------------- + ------------------------------------------------ + ---------------------------------------------------------------------- + -------------------------------------------------------- + ------------------------------------------------------- + ---------------------------------------------------------- + -------------------------------------------------------- + --------------------------------------------------------- + ---------------------------------------------------------- - ----------------------------------------------------------- - ---------------------------------------------------------- - ---------------------------------------------------------------------- + ------------------------------------------------ + ------------------------------------------------------------| for |------| > 1 | | | | -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y ________________________ -15 + 15*x + 15*y 2 2 2 -15 + 15*x + 15*y ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ -15 + 15*x + 15*y -15 + 15*x + 15*y -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y ________________________ ________________________ 2 | |-1 + y| | | | | (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x / x / x / x ___________________ / x / x ___________________ (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x ___________________ / x (-15 + 15*x + 15*y) | | | | | / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / -1 + ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / -1 + ----------------- *(-15 + 15*x + 15*y) | | | | \ \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) / |- | < dx for -1 + y > 0 | | | / _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ \ | | | | / x 5/2 / x 3/2 / x 5/2 / x 5/2 3 / x ___________________ 2 / x ___________________ / x 3/2 / x ___________________ / x 3/2 2 / x ___________________ / x 5/2 / x 3/2 2 / x 3/2 / x 3/2 / x ___________________ 2 / x ___________________| | | | | 5/2 5/2 5/2 5/2 60* / 1 - ----------------- *polar_lift (1 - y) 8* / 1 - ----------------- *polar_lift (1 - y) 3/2 6* / 1 - ----------------- *polar_lift (1 - y) 90*x* / 1 - ----------------- *polar_lift (1 - y) 90*x * / 1 - ----------------- *\/ polar_lift(1 - y) 60*x * / 1 - ----------------- *\/ polar_lift(1 - y) 24*x* / 1 - ----------------- *polar_lift (1 - y) 3/2 3 2 3/2 ___________________ 2 ___________________ 8*x* / 1 - ----------------- *\/ polar_lift(1 - y) 8*y* / 1 - ----------------- *polar_lift (1 - y) 18*x * / 1 - ----------------- *\/ polar_lift(1 - y) 60*y* / 1 - ----------------- *polar_lift (1 - y) 120*x* / 1 - ----------------- *polar_lift (1 - y) 180*x * / 1 - ----------------- *polar_lift (1 - y) 120*x*y* / 1 - ----------------- *polar_lift (1 - y) 8*x*y* / 1 - ----------------- *\/ polar_lift(1 - y) ___________________ 2 60*y*x * / 1 - ----------------- *\/ polar_lift(1 - y) | | | | |4*polar_lift (1 - y) 60*polar_lift (1 - y) 60*x*polar_lift (1 - y) 60*y*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 2*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 3*x*polar_lift (1 - y) 3*x 2*x 2*y*polar_lift (1 - y) 4*x*\/ polar_lift(1 - y) 6*x *\/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 4*x*y*\/ polar_lift(1 - y) 2*y*x \/ polar_lift(1 - y) | | | | y*|---------------------- + ----------------------- - ------------------------- - ------------------------- - --------------------------------------------------- - -------------------------------------------------- - ----------------------------------------------- + -------------------------------------------------- - ----------------------------------------------------- - ------------------------------------------------------- - ------------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------- - --------------------------------------------------------------------- - --------------------------------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------------- + ---------------------------------------------------- + ------------------------------------------------------- + ----------------------------------------------------- + ------------------------------------------------------ + ------------------------------------------------------- - -------------------------------------------------------- - ------------------------------------------------------- - ----------------------------------------------- + --------------------------------------------------------------------- + ---------------------------------------------------------| otherwise | | | | -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y _______________________ -15 + 15*x + 15*y 2 2 2 -15 + 15*x + 15*y _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ -15 + 15*x + 15*y -15 + 15*x + 15*y -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y _______________________ _______________________ 2 | | | | | (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x / x ___________________ / x ___________________ / x / x / x (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x / x ___________________ (-15 + 15*x + 15*y) | | | | | / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / 1 - ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) | | | \ \ \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) / | | | / | 1 - y < | 1 - y | / | | | | / / ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ \ | | | | / x 5/2 / x 3/2 / x 5/2 / x 5/2 3 / x ___________________ 2 / x ___________________ / x 3/2 / x ___________________ / x 3/2 2 / x ___________________ / x 5/2 / x 3/2 2 / x 3/2 / x 3/2 / x ___________________ 2 / x ___________________| | | | | 5/2 5/2 5/2 5/2 60*I* / -1 + ----------------- *polar_lift (1 - y) 8*I* / -1 + ----------------- *polar_lift (1 - y) 3/2 6*I* / -1 + ----------------- *polar_lift (1 - y) 90*I*x* / -1 + ----------------- *polar_lift (1 - y) 90*I*x * / -1 + ----------------- *\/ polar_lift(1 - y) 60*I*x * / -1 + ----------------- *\/ polar_lift(1 - y) 24*I*x* / -1 + ----------------- *polar_lift (1 - y) 2 ___________________ ___________________ 3/2 2 3/2 3 8*I*x* / -1 + ----------------- *\/ polar_lift(1 - y) 8*I*y* / -1 + ----------------- *polar_lift (1 - y) 18*I*x * / -1 + ----------------- *\/ polar_lift(1 - y) 60*I*y* / -1 + ----------------- *polar_lift (1 - y) 120*I*x* / -1 + ----------------- *polar_lift (1 - y) 180*I*x * / -1 + ----------------- *polar_lift (1 - y) 120*I*x*y* / -1 + ----------------- *polar_lift (1 - y) 8*I*x*y* / -1 + ----------------- *\/ polar_lift(1 - y) 2 ___________________ 60*I*y*x * / -1 + ----------------- *\/ polar_lift(1 - y) | | | | |4*polar_lift (1 - y) 60*polar_lift (1 - y) 60*x*polar_lift (1 - y) 60*y*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 2*I*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 6*I*x *\/ polar_lift(1 - y) 4*I*x*\/ polar_lift(1 - y) 2*I*y*polar_lift (1 - y) 2*I*x 3*I*x*polar_lift (1 - y) 3*I*x \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 2*I*y*x 4*I*x*y*\/ polar_lift(1 - y) \/ polar_lift(1 - y) | | x | | | |y*|---------------------- + ----------------------- - ------------------------- - ------------------------- - ------------------------------------------------------ - ----------------------------------------------------- + ------------------------------------------------ + ----------------------------------------------------- - -------------------------------------------------------- - ---------------------------------------------------------- - ---------------------------------------------------------- - -------------------------------------------------------- - ------------------------------------------------ - ------------------------------------------------ - ------------------------------------------------ + ---------------------------------------------------------------------- + ------------------------------------------------ + ---------------------------------------------------------------------- + -------------------------------------------------------- + ------------------------------------------------------- + ---------------------------------------------------------- + -------------------------------------------------------- + --------------------------------------------------------- + ---------------------------------------------------------- - ----------------------------------------------------------- - ---------------------------------------------------------- - ---------------------------------------------------------------------- + ------------------------------------------------ + ------------------------------------------------------------| for |------| > 1 | | | | -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y ________________________ -15 + 15*x + 15*y 2 2 2 -15 + 15*x + 15*y ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ -15 + 15*x + 15*y -15 + 15*x + 15*y -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y ________________________ ________________________ 2 | |-1 + y| | | | | (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x / x / x / x ___________________ / x / x ___________________ (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x ___________________ / x (-15 + 15*x + 15*y) | | | | | / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / -1 + ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / -1 + ----------------- *(-15 + 15*x + 15*y) | | | | \ \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) / | | < dx otherwise | | | / _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ \ | | | | / x 5/2 / x 3/2 / x 5/2 / x 5/2 3 / x ___________________ 2 / x ___________________ / x 3/2 / x ___________________ / x 3/2 2 / x ___________________ / x 5/2 / x 3/2 2 / x 3/2 / x 3/2 / x ___________________ 2 / x ___________________| | | | | 5/2 5/2 5/2 5/2 60* / 1 - ----------------- *polar_lift (1 - y) 8* / 1 - ----------------- *polar_lift (1 - y) 3/2 6* / 1 - ----------------- *polar_lift (1 - y) 90*x* / 1 - ----------------- *polar_lift (1 - y) 90*x * / 1 - ----------------- *\/ polar_lift(1 - y) 60*x * / 1 - ----------------- *\/ polar_lift(1 - y) 24*x* / 1 - ----------------- *polar_lift (1 - y) 3/2 3 2 3/2 ___________________ 2 ___________________ 8*x* / 1 - ----------------- *\/ polar_lift(1 - y) 8*y* / 1 - ----------------- *polar_lift (1 - y) 18*x * / 1 - ----------------- *\/ polar_lift(1 - y) 60*y* / 1 - ----------------- *polar_lift (1 - y) 120*x* / 1 - ----------------- *polar_lift (1 - y) 180*x * / 1 - ----------------- *polar_lift (1 - y) 120*x*y* / 1 - ----------------- *polar_lift (1 - y) 8*x*y* / 1 - ----------------- *\/ polar_lift(1 - y) ___________________ 2 60*y*x * / 1 - ----------------- *\/ polar_lift(1 - y) | | | | |4*polar_lift (1 - y) 60*polar_lift (1 - y) 60*x*polar_lift (1 - y) 60*y*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 2*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 3*x*polar_lift (1 - y) 3*x 2*x 2*y*polar_lift (1 - y) 4*x*\/ polar_lift(1 - y) 6*x *\/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 4*x*y*\/ polar_lift(1 - y) 2*y*x \/ polar_lift(1 - y) | | | | y*|---------------------- + ----------------------- - ------------------------- - ------------------------- - --------------------------------------------------- - -------------------------------------------------- - ----------------------------------------------- + -------------------------------------------------- - ----------------------------------------------------- - ------------------------------------------------------- - ------------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------- - --------------------------------------------------------------------- - --------------------------------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------------- + ---------------------------------------------------- + ------------------------------------------------------- + ----------------------------------------------------- + ------------------------------------------------------ + ------------------------------------------------------- - -------------------------------------------------------- - ------------------------------------------------------- - ----------------------------------------------- + --------------------------------------------------------------------- + ---------------------------------------------------------| otherwise | | | | -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y _______________________ -15 + 15*x + 15*y 2 2 2 -15 + 15*x + 15*y _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ -15 + 15*x + 15*y -15 + 15*x + 15*y -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y _______________________ _______________________ 2 | | | | | (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x / x ___________________ / x ___________________ / x / x / x (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x / x ___________________ (-15 + 15*x + 15*y) | | | | | / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / 1 - ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) | | | \ \ \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) / | | | / \ 0
$$\begin{cases} - \int\limits_{1 - y}^{0} \begin{cases} y \left(- \frac{90 i x^{3} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{3 i x^{3}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{60 i x^{2} y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{2 i x^{2} y}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{18 i x^{2} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} + \frac{180 i x^{2} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 i x^{2} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} + \frac{2 i x^{2}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} - \frac{8 i x y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{120 i x y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{4 i x y \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} - \frac{24 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{8 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{90 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{120 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 x \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{3 i x \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} - \frac{4 i x \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} + \frac{8 i y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 i y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 y \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{2 i y \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} + \frac{6 i \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{8 i \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{60 i \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{4 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{2 i \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)}\right) & \text{for}\: \left|{\frac{x}{y - 1}}\right| > 1 \\y \left(- \frac{90 x^{3} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{3 x^{3}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{60 x^{2} y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{2 x^{2} y}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{18 x^{2} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} + \frac{180 x^{2} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 x^{2} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{6 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} - \frac{2 x^{2}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} - \frac{8 x y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{120 x y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{4 x y \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} - \frac{24 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{8 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{90 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{120 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 x \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{3 x \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} + \frac{4 x \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} + \frac{8 y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 y \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{2 y \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} + \frac{6 \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{8 \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{60 \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{4 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{2 \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)}\right) & \text{otherwise} \end{cases}\, dx & \text{for}\: y - 1 > 0 \\\int\limits_{0}^{1 - y} \begin{cases} y \left(- \frac{90 i x^{3} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{3 i x^{3}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{60 i x^{2} y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{2 i x^{2} y}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{18 i x^{2} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} + \frac{180 i x^{2} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 i x^{2} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} + \frac{2 i x^{2}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} - \frac{8 i x y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{120 i x y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{4 i x y \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} - \frac{24 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{8 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{90 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{120 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 x \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{3 i x \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} - \frac{4 i x \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} + \frac{8 i y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 i y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 y \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{2 i y \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} + \frac{6 i \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{8 i \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{60 i \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{4 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{2 i \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)}\right) & \text{for}\: \left|{\frac{x}{y - 1}}\right| > 1 \\y \left(- \frac{90 x^{3} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{3 x^{3}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{60 x^{2} y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{2 x^{2} y}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{18 x^{2} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} + \frac{180 x^{2} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 x^{2} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{6 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} - \frac{2 x^{2}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} - \frac{8 x y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{120 x y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{4 x y \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} - \frac{24 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{8 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{90 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{120 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 x \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{3 x \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} + \frac{4 x \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} + \frac{8 y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 y \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{2 y \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} + \frac{6 \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{8 \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{60 \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{4 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{2 \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)}\right) & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
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/ 0 | / | | | | / / ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ \ | | | | / x 5/2 / x 3/2 / x 5/2 / x 5/2 3 / x ___________________ 2 / x ___________________ / x 3/2 / x ___________________ / x 3/2 2 / x ___________________ / x 5/2 / x 3/2 2 / x 3/2 / x 3/2 / x ___________________ 2 / x ___________________| | | | | 5/2 5/2 5/2 5/2 60*I* / -1 + ----------------- *polar_lift (1 - y) 8*I* / -1 + ----------------- *polar_lift (1 - y) 3/2 6*I* / -1 + ----------------- *polar_lift (1 - y) 90*I*x* / -1 + ----------------- *polar_lift (1 - y) 90*I*x * / -1 + ----------------- *\/ polar_lift(1 - y) 60*I*x * / -1 + ----------------- *\/ polar_lift(1 - y) 24*I*x* / -1 + ----------------- *polar_lift (1 - y) 2 ___________________ ___________________ 3/2 2 3/2 3 8*I*x* / -1 + ----------------- *\/ polar_lift(1 - y) 8*I*y* / -1 + ----------------- *polar_lift (1 - y) 18*I*x * / -1 + ----------------- *\/ polar_lift(1 - y) 60*I*y* / -1 + ----------------- *polar_lift (1 - y) 120*I*x* / -1 + ----------------- *polar_lift (1 - y) 180*I*x * / -1 + ----------------- *polar_lift (1 - y) 120*I*x*y* / -1 + ----------------- *polar_lift (1 - y) 8*I*x*y* / -1 + ----------------- *\/ polar_lift(1 - y) 2 ___________________ 60*I*y*x * / -1 + ----------------- *\/ polar_lift(1 - y) | | | | |4*polar_lift (1 - y) 60*polar_lift (1 - y) 60*x*polar_lift (1 - y) 60*y*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 2*I*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 6*I*x *\/ polar_lift(1 - y) 4*I*x*\/ polar_lift(1 - y) 2*I*y*polar_lift (1 - y) 2*I*x 3*I*x*polar_lift (1 - y) 3*I*x \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 2*I*y*x 4*I*x*y*\/ polar_lift(1 - y) \/ polar_lift(1 - y) | | x | | | |y*|---------------------- + ----------------------- - ------------------------- - ------------------------- - ------------------------------------------------------ - ----------------------------------------------------- + ------------------------------------------------ + ----------------------------------------------------- - -------------------------------------------------------- - ---------------------------------------------------------- - ---------------------------------------------------------- - -------------------------------------------------------- - ------------------------------------------------ - ------------------------------------------------ - ------------------------------------------------ + ---------------------------------------------------------------------- + ------------------------------------------------ + ---------------------------------------------------------------------- + -------------------------------------------------------- + ------------------------------------------------------- + ---------------------------------------------------------- + -------------------------------------------------------- + --------------------------------------------------------- + ---------------------------------------------------------- - ----------------------------------------------------------- - ---------------------------------------------------------- - ---------------------------------------------------------------------- + ------------------------------------------------ + ------------------------------------------------------------| for |------| > 1 | | | | -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y ________________________ -15 + 15*x + 15*y 2 2 2 -15 + 15*x + 15*y ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ -15 + 15*x + 15*y -15 + 15*x + 15*y -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y ________________________ ________________________ 2 | |-1 + y| | | | | (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x / x / x / x ___________________ / x / x ___________________ (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x ___________________ / x (-15 + 15*x + 15*y) | | | | | / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / -1 + ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / -1 + ----------------- *(-15 + 15*x + 15*y) | | | | \ \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) / |- | < dx for -1 + y > 0 | | | / _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ \ | | | | / x 5/2 / x 3/2 / x 5/2 / x 5/2 3 / x ___________________ 2 / x ___________________ / x 3/2 / x ___________________ / x 3/2 2 / x ___________________ / x 5/2 / x 3/2 2 / x 3/2 / x 3/2 / x ___________________ 2 / x ___________________| | | | | 5/2 5/2 5/2 5/2 60* / 1 - ----------------- *polar_lift (1 - y) 8* / 1 - ----------------- *polar_lift (1 - y) 3/2 6* / 1 - ----------------- *polar_lift (1 - y) 90*x* / 1 - ----------------- *polar_lift (1 - y) 90*x * / 1 - ----------------- *\/ polar_lift(1 - y) 60*x * / 1 - ----------------- *\/ polar_lift(1 - y) 24*x* / 1 - ----------------- *polar_lift (1 - y) 3/2 3 2 3/2 ___________________ 2 ___________________ 8*x* / 1 - ----------------- *\/ polar_lift(1 - y) 8*y* / 1 - ----------------- *polar_lift (1 - y) 18*x * / 1 - ----------------- *\/ polar_lift(1 - y) 60*y* / 1 - ----------------- *polar_lift (1 - y) 120*x* / 1 - ----------------- *polar_lift (1 - y) 180*x * / 1 - ----------------- *polar_lift (1 - y) 120*x*y* / 1 - ----------------- *polar_lift (1 - y) 8*x*y* / 1 - ----------------- *\/ polar_lift(1 - y) ___________________ 2 60*y*x * / 1 - ----------------- *\/ polar_lift(1 - y) | | | | |4*polar_lift (1 - y) 60*polar_lift (1 - y) 60*x*polar_lift (1 - y) 60*y*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 2*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 3*x*polar_lift (1 - y) 3*x 2*x 2*y*polar_lift (1 - y) 4*x*\/ polar_lift(1 - y) 6*x *\/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 4*x*y*\/ polar_lift(1 - y) 2*y*x \/ polar_lift(1 - y) | | | | y*|---------------------- + ----------------------- - ------------------------- - ------------------------- - --------------------------------------------------- - -------------------------------------------------- - ----------------------------------------------- + -------------------------------------------------- - ----------------------------------------------------- - ------------------------------------------------------- - ------------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------- - --------------------------------------------------------------------- - --------------------------------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------------- + ---------------------------------------------------- + ------------------------------------------------------- + ----------------------------------------------------- + ------------------------------------------------------ + ------------------------------------------------------- - -------------------------------------------------------- - ------------------------------------------------------- - ----------------------------------------------- + --------------------------------------------------------------------- + ---------------------------------------------------------| otherwise | | | | -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y _______________________ -15 + 15*x + 15*y 2 2 2 -15 + 15*x + 15*y _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ -15 + 15*x + 15*y -15 + 15*x + 15*y -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y _______________________ _______________________ 2 | | | | | (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x / x ___________________ / x ___________________ / x / x / x (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x / x ___________________ (-15 + 15*x + 15*y) | | | | | / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / 1 - ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) | | | \ \ \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) / | | | / | 1 - y < | 1 - y | / | | | | / / ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ \ | | | | / x 5/2 / x 3/2 / x 5/2 / x 5/2 3 / x ___________________ 2 / x ___________________ / x 3/2 / x ___________________ / x 3/2 2 / x ___________________ / x 5/2 / x 3/2 2 / x 3/2 / x 3/2 / x ___________________ 2 / x ___________________| | | | | 5/2 5/2 5/2 5/2 60*I* / -1 + ----------------- *polar_lift (1 - y) 8*I* / -1 + ----------------- *polar_lift (1 - y) 3/2 6*I* / -1 + ----------------- *polar_lift (1 - y) 90*I*x* / -1 + ----------------- *polar_lift (1 - y) 90*I*x * / -1 + ----------------- *\/ polar_lift(1 - y) 60*I*x * / -1 + ----------------- *\/ polar_lift(1 - y) 24*I*x* / -1 + ----------------- *polar_lift (1 - y) 2 ___________________ ___________________ 3/2 2 3/2 3 8*I*x* / -1 + ----------------- *\/ polar_lift(1 - y) 8*I*y* / -1 + ----------------- *polar_lift (1 - y) 18*I*x * / -1 + ----------------- *\/ polar_lift(1 - y) 60*I*y* / -1 + ----------------- *polar_lift (1 - y) 120*I*x* / -1 + ----------------- *polar_lift (1 - y) 180*I*x * / -1 + ----------------- *polar_lift (1 - y) 120*I*x*y* / -1 + ----------------- *polar_lift (1 - y) 8*I*x*y* / -1 + ----------------- *\/ polar_lift(1 - y) 2 ___________________ 60*I*y*x * / -1 + ----------------- *\/ polar_lift(1 - y) | | | | |4*polar_lift (1 - y) 60*polar_lift (1 - y) 60*x*polar_lift (1 - y) 60*y*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 2*I*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 6*I*x *\/ polar_lift(1 - y) 4*I*x*\/ polar_lift(1 - y) 2*I*y*polar_lift (1 - y) 2*I*x 3*I*x*polar_lift (1 - y) 3*I*x \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 2*I*y*x 4*I*x*y*\/ polar_lift(1 - y) \/ polar_lift(1 - y) | | x | | | |y*|---------------------- + ----------------------- - ------------------------- - ------------------------- - ------------------------------------------------------ - ----------------------------------------------------- + ------------------------------------------------ + ----------------------------------------------------- - -------------------------------------------------------- - ---------------------------------------------------------- - ---------------------------------------------------------- - -------------------------------------------------------- - ------------------------------------------------ - ------------------------------------------------ - ------------------------------------------------ + ---------------------------------------------------------------------- + ------------------------------------------------ + ---------------------------------------------------------------------- + -------------------------------------------------------- + ------------------------------------------------------- + ---------------------------------------------------------- + -------------------------------------------------------- + --------------------------------------------------------- + ---------------------------------------------------------- - ----------------------------------------------------------- - ---------------------------------------------------------- - ---------------------------------------------------------------------- + ------------------------------------------------ + ------------------------------------------------------------| for |------| > 1 | | | | -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y ________________________ -15 + 15*x + 15*y 2 2 2 -15 + 15*x + 15*y ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ -15 + 15*x + 15*y -15 + 15*x + 15*y -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y ________________________ ________________________ 2 | |-1 + y| | | | | (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x / x / x / x ___________________ / x / x ___________________ (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x ___________________ / x (-15 + 15*x + 15*y) | | | | | / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / -1 + ----------------- *(-15 + 15*x + 15*y) / -1 + ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / -1 + ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / -1 + ----------------- *(-15 + 15*x + 15*y) | | | | \ \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) / | | < dx otherwise | | | / _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ \ | | | | / x 5/2 / x 3/2 / x 5/2 / x 5/2 3 / x ___________________ 2 / x ___________________ / x 3/2 / x ___________________ / x 3/2 2 / x ___________________ / x 5/2 / x 3/2 2 / x 3/2 / x 3/2 / x ___________________ 2 / x ___________________| | | | | 5/2 5/2 5/2 5/2 60* / 1 - ----------------- *polar_lift (1 - y) 8* / 1 - ----------------- *polar_lift (1 - y) 3/2 6* / 1 - ----------------- *polar_lift (1 - y) 90*x* / 1 - ----------------- *polar_lift (1 - y) 90*x * / 1 - ----------------- *\/ polar_lift(1 - y) 60*x * / 1 - ----------------- *\/ polar_lift(1 - y) 24*x* / 1 - ----------------- *polar_lift (1 - y) 3/2 3 2 3/2 ___________________ 2 ___________________ 8*x* / 1 - ----------------- *\/ polar_lift(1 - y) 8*y* / 1 - ----------------- *polar_lift (1 - y) 18*x * / 1 - ----------------- *\/ polar_lift(1 - y) 60*y* / 1 - ----------------- *polar_lift (1 - y) 120*x* / 1 - ----------------- *polar_lift (1 - y) 180*x * / 1 - ----------------- *polar_lift (1 - y) 120*x*y* / 1 - ----------------- *polar_lift (1 - y) 8*x*y* / 1 - ----------------- *\/ polar_lift(1 - y) ___________________ 2 60*y*x * / 1 - ----------------- *\/ polar_lift(1 - y) | | | | |4*polar_lift (1 - y) 60*polar_lift (1 - y) 60*x*polar_lift (1 - y) 60*y*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 2*polar_lift (1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 3*x*polar_lift (1 - y) 3*x 2*x 2*y*polar_lift (1 - y) 4*x*\/ polar_lift(1 - y) 6*x *\/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) 4*x*y*\/ polar_lift(1 - y) 2*y*x \/ polar_lift(1 - y) | | | | y*|---------------------- + ----------------------- - ------------------------- - ------------------------- - --------------------------------------------------- - -------------------------------------------------- - ----------------------------------------------- + -------------------------------------------------- - ----------------------------------------------------- - ------------------------------------------------------- - ------------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------- - --------------------------------------------------------------------- - --------------------------------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------------- + ---------------------------------------------------- + ------------------------------------------------------- + ----------------------------------------------------- + ------------------------------------------------------ + ------------------------------------------------------- - -------------------------------------------------------- - ------------------------------------------------------- - ----------------------------------------------- + --------------------------------------------------------------------- + ---------------------------------------------------------| otherwise | | | | -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y _______________________ -15 + 15*x + 15*y 2 2 2 -15 + 15*x + 15*y _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ -15 + 15*x + 15*y -15 + 15*x + 15*y -15 + 15*x + 15*y 2 2 2 2 -15 + 15*x + 15*y _______________________ _______________________ 2 | | | | | (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x / x ___________________ / x ___________________ / x / x / x (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) (-15 + 15*x + 15*y) / x / x ___________________ (-15 + 15*x + 15*y) | | | | | / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / 1 - ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y) / 1 - ----------------- *(-15 + 15*x + 15*y)*\/ polar_lift(1 - y) | | | \ \ \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) \/ polar_lift(1 - y) / | | | / \ 0
$$\begin{cases} - \int\limits_{1 - y}^{0} \begin{cases} y \left(- \frac{90 i x^{3} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{3 i x^{3}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{60 i x^{2} y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{2 i x^{2} y}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{18 i x^{2} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} + \frac{180 i x^{2} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 i x^{2} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} + \frac{2 i x^{2}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} - \frac{8 i x y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{120 i x y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{4 i x y \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} - \frac{24 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{8 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{90 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{120 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 x \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{3 i x \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} - \frac{4 i x \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} + \frac{8 i y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 i y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 y \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{2 i y \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} + \frac{6 i \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{8 i \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{60 i \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{4 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{2 i \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)}\right) & \text{for}\: \left|{\frac{x}{y - 1}}\right| > 1 \\y \left(- \frac{90 x^{3} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{3 x^{3}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{60 x^{2} y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{2 x^{2} y}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{18 x^{2} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} + \frac{180 x^{2} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 x^{2} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{6 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} - \frac{2 x^{2}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} - \frac{8 x y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{120 x y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{4 x y \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} - \frac{24 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{8 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{90 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{120 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 x \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{3 x \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} + \frac{4 x \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} + \frac{8 y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 y \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{2 y \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} + \frac{6 \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{8 \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{60 \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{4 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{2 \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)}\right) & \text{otherwise} \end{cases}\, dx & \text{for}\: y - 1 > 0 \\\int\limits_{0}^{1 - y} \begin{cases} y \left(- \frac{90 i x^{3} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{3 i x^{3}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{60 i x^{2} y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{2 i x^{2} y}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{18 i x^{2} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} + \frac{180 i x^{2} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 i x^{2} \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} + \frac{2 i x^{2}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} - \frac{8 i x y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{120 i x y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{4 i x y \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} - \frac{24 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{8 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{90 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{120 i x \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 x \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{3 i x \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} - \frac{4 i x \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} + \frac{8 i y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 i y \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 y \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{2 i y \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)} + \frac{6 i \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{8 i \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{60 i \sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{4 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{2 i \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{\frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} - 1} \left(15 x + 15 y - 15\right)}\right) & \text{for}\: \left|{\frac{x}{y - 1}}\right| > 1 \\y \left(- \frac{90 x^{3} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{3 x^{3}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{60 x^{2} y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{2 x^{2} y}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} + \frac{18 x^{2} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} + \frac{180 x^{2} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 x^{2} \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{6 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} - \frac{2 x^{2}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}} - \frac{8 x y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{120 x y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{4 x y \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} - \frac{24 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{8 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{15 x + 15 y - 15} - \frac{90 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{120 x \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 x \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{3 x \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} + \frac{4 x \sqrt{\operatorname{polar\_lift}{\left(1 - y \right)}}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} + \frac{8 y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 y \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{60 y \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{2 y \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)} + \frac{6 \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{8 \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} - \frac{60 \sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} + \frac{4 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{15 x + 15 y - 15} + \frac{60 \operatorname{polar\_lift}^{\frac{5}{2}}{\left(1 - y \right)}}{\left(15 x + 15 y - 15\right)^{2}} - \frac{2 \operatorname{polar\_lift}^{\frac{3}{2}}{\left(1 - y \right)}}{\sqrt{- \frac{x}{\operatorname{polar\_lift}{\left(1 - y \right)}} + 1} \left(15 x + 15 y - 15\right)}\right) & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
Piecewise((-Integral(Piecewise((y*(4*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y) + 60*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 60*x*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 60*y*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 60*i*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 8*i*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y) + 2*i*polar_lift(1 - y)^(3/2)/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 6*i*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y) - 90*i*x*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 90*i*x^3*sqrt(-1 + x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y)^2 - 60*i*x^2*sqrt(-1 + x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y)^2 - 24*i*x*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y) - 6*i*x^2*sqrt(polar_lift(1 - y))/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) - 4*i*x*sqrt(polar_lift(1 - y))/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) - 2*i*y*polar_lift(1 - y)^(3/2)/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 2*i*x^2/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)*sqrt(polar_lift(1 - y))) + 3*i*x*polar_lift(1 - y)^(3/2)/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 3*i*x^3/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)*sqrt(polar_lift(1 - y))) + 8*i*x*sqrt(-1 + x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y) + 8*i*y*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y) + 18*i*x^2*sqrt(-1 + x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y) + 60*i*y*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 + 120*i*x*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y)^2 + 180*i*x^2*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y)^2 - 120*i*x*y*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y)^2 - 8*i*x*y*sqrt(-1 + x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y) - 2*i*y*x^2/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)*sqrt(polar_lift(1 - y))) + 4*i*x*y*sqrt(polar_lift(1 - y))/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 60*i*y*x^2*sqrt(-1 + x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y)^2), Abs(x/(-1 + y)) > 1), (y*(4*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y) + 60*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 60*x*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 60*y*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 60*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 8*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y) - 2*polar_lift(1 - y)^(3/2)/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 6*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y) - 90*x*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 90*x^3*sqrt(1 - x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y)^2 - 60*x^2*sqrt(1 - x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y)^2 - 24*x*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y) - 3*x*polar_lift(1 - y)^(3/2)/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) - 3*x^3/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)*sqrt(polar_lift(1 - y))) - 2*x^2/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)*sqrt(polar_lift(1 - y))) + 2*y*polar_lift(1 - y)^(3/2)/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 4*x*sqrt(polar_lift(1 - y))/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 6*x^2*sqrt(polar_lift(1 - y))/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 8*x*sqrt(1 - x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y) + 8*y*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y) + 18*x^2*sqrt(1 - x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y) + 60*y*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 + 120*x*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y)^2 + 180*x^2*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y)^2 - 120*x*y*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y)^2 - 8*x*y*sqrt(1 - x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y) - 4*x*y*sqrt(polar_lift(1 - y))/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 2*y*x^2/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)*sqrt(polar_lift(1 - y))) + 60*y*x^2*sqrt(1 - x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y)^2), True)), (x, 1 - y, 0)), -1 + y > 0), (Integral(Piecewise((y*(4*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y) + 60*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 60*x*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 60*y*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 60*i*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 8*i*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y) + 2*i*polar_lift(1 - y)^(3/2)/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 6*i*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y) - 90*i*x*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 90*i*x^3*sqrt(-1 + x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y)^2 - 60*i*x^2*sqrt(-1 + x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y)^2 - 24*i*x*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y) - 6*i*x^2*sqrt(polar_lift(1 - y))/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) - 4*i*x*sqrt(polar_lift(1 - y))/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) - 2*i*y*polar_lift(1 - y)^(3/2)/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 2*i*x^2/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)*sqrt(polar_lift(1 - y))) + 3*i*x*polar_lift(1 - y)^(3/2)/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 3*i*x^3/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)*sqrt(polar_lift(1 - y))) + 8*i*x*sqrt(-1 + x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y) + 8*i*y*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y) + 18*i*x^2*sqrt(-1 + x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y) + 60*i*y*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 + 120*i*x*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y)^2 + 180*i*x^2*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y)^2 - 120*i*x*y*sqrt(-1 + x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y)^2 - 8*i*x*y*sqrt(-1 + x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y) - 2*i*y*x^2/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)*sqrt(polar_lift(1 - y))) + 4*i*x*y*sqrt(polar_lift(1 - y))/(sqrt(-1 + x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 60*i*y*x^2*sqrt(-1 + x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y)^2), Abs(x/(-1 + y)) > 1), (y*(4*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y) + 60*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 60*x*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 60*y*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 60*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 8*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y) - 2*polar_lift(1 - y)^(3/2)/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 6*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y) - 90*x*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 - 90*x^3*sqrt(1 - x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y)^2 - 60*x^2*sqrt(1 - x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y)^2 - 24*x*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y) - 3*x*polar_lift(1 - y)^(3/2)/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) - 3*x^3/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)*sqrt(polar_lift(1 - y))) - 2*x^2/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)*sqrt(polar_lift(1 - y))) + 2*y*polar_lift(1 - y)^(3/2)/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 4*x*sqrt(polar_lift(1 - y))/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 6*x^2*sqrt(polar_lift(1 - y))/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 8*x*sqrt(1 - x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y) + 8*y*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y) + 18*x^2*sqrt(1 - x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y) + 60*y*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(5/2)/(-15 + 15*x + 15*y)^2 + 120*x*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y)^2 + 180*x^2*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y)^2 - 120*x*y*sqrt(1 - x/polar_lift(1 - y))*polar_lift(1 - y)^(3/2)/(-15 + 15*x + 15*y)^2 - 8*x*y*sqrt(1 - x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y) - 4*x*y*sqrt(polar_lift(1 - y))/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)) + 2*y*x^2/(sqrt(1 - x/polar_lift(1 - y))*(-15 + 15*x + 15*y)*sqrt(polar_lift(1 - y))) + 60*y*x^2*sqrt(1 - x/polar_lift(1 - y))*sqrt(polar_lift(1 - y))/(-15 + 15*x + 15*y)^2), True)), (x, 0, 1 - y)), True))
Use the examples entering the upper and lower limits of integration.