3.273 \(\int x \sin ^{-1}(x) \, dx\)

Optimal. Leaf size=32 \[ \frac{1}{4} \sqrt{1-x^2} x+\frac{1}{2} x^2 \sin ^{-1}(x)-\frac{1}{4} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0267451, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75 \[ \frac{1}{4} \sqrt{1-x^2} x+\frac{1}{2} x^2 \sin ^{-1}(x)-\frac{1}{4} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[x*ArcSin[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.03337, size = 24, normalized size = 0.75 \[ \frac{x^{2} \operatorname{asin}{\left (x \right )}}{2} + \frac{x \sqrt{- x^{2} + 1}}{4} - \frac{\operatorname{asin}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*asin(x),x)

[Out]

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

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Mathematica [A]  time = 0.0152347, size = 28, normalized size = 0.88 \[ \frac{1}{4} \left (\sqrt{1-x^2} x+\left (2 x^2-1\right ) \sin ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*ArcSin[x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 25, normalized size = 0.8 \[ -{\frac{\arcsin \left ( x \right ) }{4}}+{\frac{{x}^{2}\arcsin \left ( x \right ) }{2}}+{\frac{x}{4}\sqrt{-{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsin(x)*x,x)

[Out]

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

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Maxima [A]  time = 1.48113, size = 32, normalized size = 1. \[ \frac{1}{2} \, x^{2} \arcsin \left (x\right ) + \frac{1}{4} \, \sqrt{-x^{2} + 1} x - \frac{1}{4} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.220367, size = 32, normalized size = 1. \[ \frac{1}{4} \,{\left (2 \, x^{2} - 1\right )} \arcsin \left (x\right ) + \frac{1}{4} \, \sqrt{-x^{2} + 1} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.247828, size = 24, normalized size = 0.75 \[ \frac{x^{2} \operatorname{asin}{\left (x \right )}}{2} + \frac{x \sqrt{- x^{2} + 1}}{4} - \frac{\operatorname{asin}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*asin(x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.227115, size = 35, normalized size = 1.09 \[ \frac{1}{2} \,{\left (x^{2} - 1\right )} \arcsin \left (x\right ) + \frac{1}{4} \, \sqrt{-x^{2} + 1} x + \frac{1}{4} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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