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

Optimal. Leaf size=17 \[ x-\sqrt{1-x^2} \sin ^{-1}(x) \]

[Out]

x - Sqrt[1 - x^2]*ArcSin[x]

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Rubi [A]  time = 0.0480883, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ x-\sqrt{1-x^2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

x - Sqrt[1 - x^2]*ArcSin[x]

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Rubi in Sympy [A]  time = 4.36991, size = 12, normalized size = 0.71 \[ x - \sqrt{- x^{2} + 1} \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - sqrt(-x**2 + 1)*asin(x)

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Mathematica [A]  time = 0.0153019, size = 17, normalized size = 1. \[ x-\sqrt{1-x^2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

x - Sqrt[1 - x^2]*ArcSin[x]

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Maple [A]  time = 0.069, size = 16, normalized size = 0.9 \[ x-\arcsin \left ( x \right ) \sqrt{-{x}^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.60183, size = 20, normalized size = 1.18 \[ -\sqrt{-x^{2} + 1} \arcsin \left (x\right ) + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1)*arcsin(x) + x

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Fricas [A]  time = 0.236311, size = 20, normalized size = 1.18 \[ -\sqrt{-x^{2} + 1} \arcsin \left (x\right ) + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1)*arcsin(x) + x

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Sympy [A]  time = 0.225616, size = 12, normalized size = 0.71 \[ x - \sqrt{- x^{2} + 1} \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - sqrt(-x**2 + 1)*asin(x)

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GIAC/XCAS [A]  time = 0.20658, size = 20, normalized size = 1.18 \[ -\sqrt{-x^{2} + 1} \arcsin \left (x\right ) + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1)*arcsin(x) + x