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

Optimal. Leaf size=16 \[ \sqrt{1-x^2}+x \sin ^{-1}(x) \]

[Out]

Sqrt[1 - x^2] + x*ArcSin[x]

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Rubi [A]  time = 0.00831988, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1. \[ \sqrt{1-x^2}+x \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[ArcSin[x],x]

[Out]

Sqrt[1 - x^2] + x*ArcSin[x]

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Rubi in Sympy [A]  time = 1.062, size = 12, normalized size = 0.75 \[ x \operatorname{asin}{\left (x \right )} + \sqrt{- x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(asin(x),x)

[Out]

x*asin(x) + sqrt(-x**2 + 1)

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Mathematica [A]  time = 0.00451368, size = 16, normalized size = 1. \[ \sqrt{1-x^2}+x \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[ArcSin[x],x]

[Out]

Sqrt[1 - x^2] + x*ArcSin[x]

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Maple [A]  time = 0., size = 15, normalized size = 0.9 \[ \arcsin \left ( x \right ) x+\sqrt{-{x}^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsin(x),x)

[Out]

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

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Maxima [A]  time = 1.50196, size = 19, normalized size = 1.19 \[ x \arcsin \left (x\right ) + \sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*arcsin(x) + sqrt(-x^2 + 1)

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Fricas [A]  time = 0.23218, size = 19, normalized size = 1.19 \[ x \arcsin \left (x\right ) + \sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*arcsin(x) + sqrt(-x^2 + 1)

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Sympy [A]  time = 0.132064, size = 12, normalized size = 0.75 \[ x \operatorname{asin}{\left (x \right )} + \sqrt{- x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(asin(x),x)

[Out]

x*asin(x) + sqrt(-x**2 + 1)

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GIAC/XCAS [A]  time = 0.201503, size = 19, normalized size = 1.19 \[ x \arcsin \left (x\right ) + \sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*arcsin(x) + sqrt(-x^2 + 1)