3.668 \(\int \frac{x^2 \sin ^{-1}(x)^3}{\sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=73 \[ -\frac{3 x^2}{8}-\frac{1}{2} x \sqrt{1-x^2} \sin ^{-1}(x)^3+\frac{3}{4} x^2 \sin ^{-1}(x)^2+\frac{3}{4} x \sqrt{1-x^2} \sin ^{-1}(x)+\frac{1}{8} \sin ^{-1}(x)^4-\frac{3}{8} \sin ^{-1}(x)^2 \]

[Out]

(-3*x^2)/8 + (3*x*Sqrt[1 - x^2]*ArcSin[x])/4 - (3*ArcSin[x]^2)/8 + (3*x^2*ArcSin
[x]^2)/4 - (x*Sqrt[1 - x^2]*ArcSin[x]^3)/2 + ArcSin[x]^4/8

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Rubi [A]  time = 0.272341, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{3 x^2}{8}-\frac{1}{2} x \sqrt{1-x^2} \sin ^{-1}(x)^3+\frac{3}{4} x^2 \sin ^{-1}(x)^2+\frac{3}{4} x \sqrt{1-x^2} \sin ^{-1}(x)+\frac{1}{8} \sin ^{-1}(x)^4-\frac{3}{8} \sin ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]  Int[(x^2*ArcSin[x]^3)/Sqrt[1 - x^2],x]

[Out]

(-3*x^2)/8 + (3*x*Sqrt[1 - x^2]*ArcSin[x])/4 - (3*ArcSin[x]^2)/8 + (3*x^2*ArcSin
[x]^2)/4 - (x*Sqrt[1 - x^2]*ArcSin[x]^3)/2 + ArcSin[x]^4/8

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 x^{2} \operatorname{asin}^{2}{\left (x \right )}}{4} - \frac{x \sqrt{- x^{2} + 1} \operatorname{asin}^{3}{\left (x \right )}}{2} + \frac{3 x \sqrt{- x^{2} + 1} \operatorname{asin}{\left (x \right )}}{4} + \frac{\operatorname{asin}^{4}{\left (x \right )}}{8} - \frac{3 \operatorname{asin}^{2}{\left (x \right )}}{8} - \frac{3 \int x\, dx}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*asin(x)**3/(-x**2+1)**(1/2),x)

[Out]

3*x**2*asin(x)**2/4 - x*sqrt(-x**2 + 1)*asin(x)**3/2 + 3*x*sqrt(-x**2 + 1)*asin(
x)/4 + asin(x)**4/8 - 3*asin(x)**2/8 - 3*Integral(x, x)/4

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Mathematica [A]  time = 0.0377257, size = 60, normalized size = 0.82 \[ \frac{1}{8} \left (-3 x^2-4 x \sqrt{1-x^2} \sin ^{-1}(x)^3+\left (6 x^2-3\right ) \sin ^{-1}(x)^2+6 x \sqrt{1-x^2} \sin ^{-1}(x)+\sin ^{-1}(x)^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*ArcSin[x]^3)/Sqrt[1 - x^2],x]

[Out]

(-3*x^2 + 6*x*Sqrt[1 - x^2]*ArcSin[x] + (-3 + 6*x^2)*ArcSin[x]^2 - 4*x*Sqrt[1 -
x^2]*ArcSin[x]^3 + ArcSin[x]^4)/8

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Maple [A]  time = 0.089, size = 69, normalized size = 1. \[{\frac{ \left ( \arcsin \left ( x \right ) \right ) ^{3}}{2} \left ( -x\sqrt{-{x}^{2}+1}+\arcsin \left ( x \right ) \right ) }+{\frac{3\, \left ( \arcsin \left ( x \right ) \right ) ^{2} \left ({x}^{2}-1 \right ) }{4}}+{\frac{3\,\arcsin \left ( x \right ) }{4} \left ( x\sqrt{-{x}^{2}+1}+\arcsin \left ( x \right ) \right ) }-{\frac{3\, \left ( \arcsin \left ( x \right ) \right ) ^{2}}{8}}-{\frac{3\,{x}^{2}}{8}}-{\frac{3\, \left ( \arcsin \left ( x \right ) \right ) ^{4}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*arcsin(x)^3/(-x^2+1)^(1/2),x)

[Out]

1/2*arcsin(x)^3*(-x*(-x^2+1)^(1/2)+arcsin(x))+3/4*arcsin(x)^2*(x^2-1)+3/4*arcsin
(x)*(x*(-x^2+1)^(1/2)+arcsin(x))-3/8*arcsin(x)^2-3/8*x^2-3/8*arcsin(x)^4

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Maxima [A]  time = 10.2824, size = 194, normalized size = 2.66 \[ -\frac{1}{2} \,{\left (\sqrt{-x^{2} + 1} x - \arcsin \left (x\right )\right )} \arcsin \left (x\right )^{3} - \frac{1}{8} \, \arctan \left (x, \sqrt{x + 1} \sqrt{-x + 1}\right )^{4} + \frac{3}{4} \,{\left (x^{2} - \arctan \left (x, \sqrt{x + 1} \sqrt{-x + 1}\right )^{2}\right )} \arcsin \left (x\right )^{2} - \frac{3}{8} \, x^{2} + \frac{1}{4} \,{\left (2 \, \arctan \left (x, \sqrt{x + 1} \sqrt{-x + 1}\right )^{3} + 3 \, \sqrt{-x^{2} + 1} x - 3 \, \arctan \left (x, \sqrt{-x^{2} + 1}\right )\right )} \arcsin \left (x\right ) + \frac{3}{8} \, \arctan \left (x, \sqrt{x + 1} \sqrt{-x + 1}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*arcsin(x)^3/sqrt(-x^2 + 1),x, algorithm="maxima")

[Out]

-1/2*(sqrt(-x^2 + 1)*x - arcsin(x))*arcsin(x)^3 - 1/8*arctan2(x, sqrt(x + 1)*sqr
t(-x + 1))^4 + 3/4*(x^2 - arctan2(x, sqrt(x + 1)*sqrt(-x + 1))^2)*arcsin(x)^2 -
3/8*x^2 + 1/4*(2*arctan2(x, sqrt(x + 1)*sqrt(-x + 1))^3 + 3*sqrt(-x^2 + 1)*x - 3
*arctan2(x, sqrt(-x^2 + 1)))*arcsin(x) + 3/8*arctan2(x, sqrt(x + 1)*sqrt(-x + 1)
)^2

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Fricas [A]  time = 0.254375, size = 66, normalized size = 0.9 \[ \frac{1}{8} \, \arcsin \left (x\right )^{4} + \frac{3}{8} \,{\left (2 \, x^{2} - 1\right )} \arcsin \left (x\right )^{2} - \frac{3}{8} \, x^{2} - \frac{1}{4} \,{\left (2 \, x \arcsin \left (x\right )^{3} - 3 \, x \arcsin \left (x\right )\right )} \sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*arcsin(x)^3/sqrt(-x^2 + 1),x, algorithm="fricas")

[Out]

1/8*arcsin(x)^4 + 3/8*(2*x^2 - 1)*arcsin(x)^2 - 3/8*x^2 - 1/4*(2*x*arcsin(x)^3 -
3*x*arcsin(x))*sqrt(-x^2 + 1)

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Sympy [A]  time = 1.83688, size = 66, normalized size = 0.9 \[ \frac{3 x^{2} \operatorname{asin}^{2}{\left (x \right )}}{4} - \frac{3 x^{2}}{8} - \frac{x \sqrt{- x^{2} + 1} \operatorname{asin}^{3}{\left (x \right )}}{2} + \frac{3 x \sqrt{- x^{2} + 1} \operatorname{asin}{\left (x \right )}}{4} + \frac{\operatorname{asin}^{4}{\left (x \right )}}{8} - \frac{3 \operatorname{asin}^{2}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*asin(x)**3/(-x**2+1)**(1/2),x)

[Out]

3*x**2*asin(x)**2/4 - 3*x**2/8 - x*sqrt(-x**2 + 1)*asin(x)**3/2 + 3*x*sqrt(-x**2
+ 1)*asin(x)/4 + asin(x)**4/8 - 3*asin(x)**2/8

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GIAC/XCAS [A]  time = 0.230822, size = 81, normalized size = 1.11 \[ -\frac{1}{2} \, \sqrt{-x^{2} + 1} x \arcsin \left (x\right )^{3} + \frac{1}{8} \, \arcsin \left (x\right )^{4} + \frac{3}{4} \,{\left (x^{2} - 1\right )} \arcsin \left (x\right )^{2} + \frac{3}{4} \, \sqrt{-x^{2} + 1} x \arcsin \left (x\right ) - \frac{3}{8} \, x^{2} + \frac{3}{8} \, \arcsin \left (x\right )^{2} + \frac{3}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*arcsin(x)^3/sqrt(-x^2 + 1),x, algorithm="giac")

[Out]

-1/2*sqrt(-x^2 + 1)*x*arcsin(x)^3 + 1/8*arcsin(x)^4 + 3/4*(x^2 - 1)*arcsin(x)^2
+ 3/4*sqrt(-x^2 + 1)*x*arcsin(x) - 3/8*x^2 + 3/8*arcsin(x)^2 + 3/16