3.663 \(\int \frac{\sin ^{-1}(x)}{\left (1-x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{1}{6 \left (1-x^2\right )}+\frac{1}{3} \log \left (1-x^2\right )+\frac{2 x \sin ^{-1}(x)}{3 \sqrt{1-x^2}}+\frac{x \sin ^{-1}(x)}{3 \left (1-x^2\right )^{3/2}} \]

[Out]

-1/(6*(1 - x^2)) + (x*ArcSin[x])/(3*(1 - x^2)^(3/2)) + (2*x*ArcSin[x])/(3*Sqrt[1
- x^2]) + Log[1 - x^2]/3

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Rubi [A]  time = 0.0614553, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{1}{6 \left (1-x^2\right )}+\frac{1}{3} \log \left (1-x^2\right )+\frac{2 x \sin ^{-1}(x)}{3 \sqrt{1-x^2}}+\frac{x \sin ^{-1}(x)}{3 \left (1-x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[ArcSin[x]/(1 - x^2)^(5/2),x]

[Out]

-1/(6*(1 - x^2)) + (x*ArcSin[x])/(3*(1 - x^2)^(3/2)) + (2*x*ArcSin[x])/(3*Sqrt[1
- x^2]) + Log[1 - x^2]/3

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Rubi in Sympy [A]  time = 4.91892, size = 49, normalized size = 0.79 \[ \frac{2 x \operatorname{asin}{\left (x \right )}}{3 \sqrt{- x^{2} + 1}} + \frac{x \operatorname{asin}{\left (x \right )}}{3 \left (- x^{2} + 1\right )^{\frac{3}{2}}} + \frac{\log{\left (\left (x - 1\right ) \left (x + 1\right ) \right )}}{3} - \frac{1}{6 \left (- x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(asin(x)/(-x**2+1)**(5/2),x)

[Out]

2*x*asin(x)/(3*sqrt(-x**2 + 1)) + x*asin(x)/(3*(-x**2 + 1)**(3/2)) + log((x - 1)
*(x + 1))/3 - 1/(6*(-x**2 + 1))

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Mathematica [A]  time = 0.0931762, size = 45, normalized size = 0.73 \[ \frac{1}{6} \left (\frac{1}{x^2-1}+2 \log \left (1-x^2\right )-\frac{2 x \left (2 x^2-3\right ) \sin ^{-1}(x)}{\left (1-x^2\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[ArcSin[x]/(1 - x^2)^(5/2),x]

[Out]

((-1 + x^2)^(-1) - (2*x*(-3 + 2*x^2)*ArcSin[x])/(1 - x^2)^(3/2) + 2*Log[1 - x^2]
)/6

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Maple [A]  time = 0.059, size = 63, normalized size = 1. \[{\frac{\arcsin \left ( x \right ) x}{3\, \left ({x}^{2}-1 \right ) ^{2}}\sqrt{-{x}^{2}+1}}+{\frac{1}{6\,{x}^{2}-6}}-{\frac{2\,\arcsin \left ( x \right ) x}{3\,{x}^{2}-3}\sqrt{-{x}^{2}+1}}+{\frac{\ln \left ( -{x}^{2}+1 \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsin(x)/(-x^2+1)^(5/2),x)

[Out]

1/3*x*arcsin(x)*(-x^2+1)^(1/2)/(x^2-1)^2+1/6/(x^2-1)-2/3*(-x^2+1)^(1/2)/(x^2-1)*
arcsin(x)*x+1/3*ln(-x^2+1)

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Maxima [A]  time = 1.51415, size = 65, normalized size = 1.05 \[ \frac{1}{3} \,{\left (\frac{2 \, x}{\sqrt{-x^{2} + 1}} + \frac{x}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}\right )} \arcsin \left (x\right ) + \frac{1}{6 \,{\left (x^{2} - 1\right )}} + \frac{1}{3} \, \log \left (-3 \, x^{2} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(x)/(-x^2 + 1)^(5/2),x, algorithm="maxima")

[Out]

1/3*(2*x/sqrt(-x^2 + 1) + x/(-x^2 + 1)^(3/2))*arcsin(x) + 1/6/(x^2 - 1) + 1/3*lo
g(-3*x^2 + 3)

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Fricas [A]  time = 0.225259, size = 82, normalized size = 1.32 \[ -\frac{2 \,{\left (2 \, x^{3} - 3 \, x\right )} \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - x^{2} - 2 \,{\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x^{2} - 1\right ) + 1}{6 \,{\left (x^{4} - 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(x)/(-x^2 + 1)^(5/2),x, algorithm="fricas")

[Out]

-1/6*(2*(2*x^3 - 3*x)*sqrt(-x^2 + 1)*arcsin(x) - x^2 - 2*(x^4 - 2*x^2 + 1)*log(x
^2 - 1) + 1)/(x^4 - 2*x^2 + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(asin(x)/(-x**2+1)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22125, size = 73, normalized size = 1.18 \[ -\frac{{\left (2 \, x^{2} - 3\right )} \sqrt{-x^{2} + 1} x \arcsin \left (x\right )}{3 \,{\left (x^{2} - 1\right )}^{2}} - \frac{2 \, x^{2} - 3}{6 \,{\left (x^{2} - 1\right )}} + \frac{1}{3} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(x)/(-x^2 + 1)^(5/2),x, algorithm="giac")

[Out]

-1/3*(2*x^2 - 3)*sqrt(-x^2 + 1)*x*arcsin(x)/(x^2 - 1)^2 - 1/6*(2*x^2 - 3)/(x^2 -
1) + 1/3*ln(abs(x^2 - 1))