3.350 \(\int x^3 \sin ^{-1}\left (x^2\right ) \, dx\)

Optimal. Leaf size=38 \[ -\frac{1}{8} \sin ^{-1}\left (x^2\right )+\frac{1}{8} \sqrt{1-x^4} x^2+\frac{1}{4} x^4 \sin ^{-1}\left (x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.0545805, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625 \[ -\frac{1}{8} \sin ^{-1}\left (x^2\right )+\frac{1}{8} \sqrt{1-x^4} x^2+\frac{1}{4} x^4 \sin ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^3*ArcSin[x^2],x]

[Out]

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

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*asin(x**2),x)

[Out]

Exception raised: ValueError

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Mathematica [A]  time = 0.0167284, size = 32, normalized size = 0.84 \[ \frac{1}{8} \left (\sqrt{1-x^4} x^2+\left (2 x^4-1\right ) \sin ^{-1}\left (x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*ArcSin[x^2],x]

[Out]

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

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Maple [A]  time = 0.004, size = 31, normalized size = 0.8 \[ -{\frac{\arcsin \left ({x}^{2} \right ) }{8}}+{\frac{{x}^{4}\arcsin \left ({x}^{2} \right ) }{4}}+{\frac{{x}^{2}}{8}\sqrt{-{x}^{4}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*arcsin(x^2),x)

[Out]

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

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Maxima [A]  time = 1.60733, size = 72, normalized size = 1.89 \[ \frac{1}{4} \, x^{4} \arcsin \left (x^{2}\right ) - \frac{\sqrt{-x^{4} + 1}}{8 \, x^{2}{\left (\frac{x^{4} - 1}{x^{4}} - 1\right )}} + \frac{1}{8} \, \arctan \left (\frac{\sqrt{-x^{4} + 1}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*arcsin(x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.226749, size = 38, normalized size = 1. \[ \frac{1}{8} \, \sqrt{-x^{4} + 1} x^{2} + \frac{1}{8} \,{\left (2 \, x^{4} - 1\right )} \arcsin \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*arcsin(x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.896623, size = 29, normalized size = 0.76 \[ \frac{x^{4} \operatorname{asin}{\left (x^{2} \right )}}{4} + \frac{x^{2} \sqrt{- x^{4} + 1}}{8} - \frac{\operatorname{asin}{\left (x^{2} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*asin(x**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.204829, size = 43, normalized size = 1.13 \[ \frac{1}{8} \, \sqrt{-x^{4} + 1} x^{2} + \frac{1}{4} \,{\left (x^{4} - 1\right )} \arcsin \left (x^{2}\right ) + \frac{1}{8} \, \arcsin \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*arcsin(x^2),x, algorithm="giac")

[Out]

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