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

Optimal. Leaf size=28 \[ -\frac{\sqrt{x^2} \sin ^{-1}\left (\sqrt{1-x^2}\right )^2}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.0541507, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\sqrt{x^2} \sin ^{-1}\left (\sqrt{1-x^2}\right )^2}{2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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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(asin((-x**2+1)**(1/2))/(-x**2+1)**(1/2),x)

[Out]

Exception raised: ValueError

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Mathematica [A]  time = 0.0118858, size = 28, normalized size = 1. \[ -\frac{\sqrt{x^2} \sin ^{-1}\left (\sqrt{1-x^2}\right )^2}{2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{1\arcsin \left ( \sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(arcsin((-x^2+1)^(1/2))/(-x^2+1)^(1/2),x)

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Maxima [A]  time = 1.54754, size = 27, normalized size = 0.96 \[ \arcsin \left (\sqrt{-x^{2} + 1}\right ) \arcsin \left (x\right ) + \frac{1}{2} \, \arcsin \left (x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222461, size = 19, normalized size = 0.68 \[ -\frac{1}{2} \, \arcsin \left (\sqrt{-x^{2} + 1}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*arcsin(sqrt(-x^2 + 1))^2

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Sympy [A]  time = 4.28677, size = 22, normalized size = 0.79 \[ - \frac{\sqrt{x^{2}} \operatorname{asin}^{2}{\left (\sqrt{- x^{2} + 1} \right )}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\arcsin \left (\sqrt{-x^{2} + 1}\right )}{\sqrt{-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(arcsin(sqrt(-x^2 + 1))/sqrt(-x^2 + 1), x)