3.85 \(\int \sin ^{-1}(x)^2 \, dx\)

Optimal. Leaf size=25 \[ 2 \sqrt{1-x^2} \sin ^{-1}(x)-2 x+x \sin ^{-1}(x)^2 \]

[Out]

-2*x + 2*Sqrt[1 - x^2]*ArcSin[x] + x*ArcSin[x]^2

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Rubi [A]  time = 0.0539379, antiderivative size = 25, 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 \[ 2 \sqrt{1-x^2} \sin ^{-1}(x)-2 x+x \sin ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]  Int[ArcSin[x]^2,x]

[Out]

-2*x + 2*Sqrt[1 - x^2]*ArcSin[x] + x*ArcSin[x]^2

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Rubi in Sympy [A]  time = 2.87102, size = 22, normalized size = 0.88 \[ x \operatorname{asin}^{2}{\left (x \right )} - 2 x + 2 \sqrt{- x^{2} + 1} \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(asin(x)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0117959, size = 25, normalized size = 1. \[ 2 \sqrt{1-x^2} \sin ^{-1}(x)-2 x+x \sin ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]  Integrate[ArcSin[x]^2,x]

[Out]

-2*x + 2*Sqrt[1 - x^2]*ArcSin[x] + x*ArcSin[x]^2

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Maple [A]  time = 0.158, size = 24, normalized size = 1. \[ -2\,x+x \left ( \arcsin \left ( x \right ) \right ) ^{2}+2\,\arcsin \left ( x \right ) \sqrt{-{x}^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsin(x)^2,x)

[Out]

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

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Maxima [A]  time = 1.57379, size = 31, normalized size = 1.24 \[ x \arcsin \left (x\right )^{2} + 2 \, \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219299, size = 31, normalized size = 1.24 \[ x \arcsin \left (x\right )^{2} + 2 \, \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.210235, size = 22, normalized size = 0.88 \[ x \operatorname{asin}^{2}{\left (x \right )} - 2 x + 2 \sqrt{- x^{2} + 1} \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(asin(x)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.235074, size = 31, normalized size = 1.24 \[ x \arcsin \left (x\right )^{2} + 2 \, \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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