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

Optimal. Leaf size=3 \[ \tan ^{-1}\left (\sin ^{-1}(x)\right ) \]

[Out]

ArcTan[ArcSin[x]]

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Rubi [A]  time = 0.0746184, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \tan ^{-1}\left (\sin ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[ArcSin[x]]

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Rubi in Sympy [A]  time = 4.84677, size = 3, normalized size = 1. \[ \operatorname{atan}{\left (\operatorname{asin}{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(asin(x))

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Mathematica [A]  time = 0.00707962, size = 3, normalized size = 1. \[ \tan ^{-1}\left (\sin ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[ArcSin[x]]

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Maple [A]  time = 0.009, size = 4, normalized size = 1.3 \[ \arctan \left ( \arcsin \left ( x \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(arcsin(x))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.230431, size = 4, normalized size = 1.33 \[ \arctan \left (\arcsin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(arcsin(x))

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Sympy [A]  time = 0.714087, size = 3, normalized size = 1. \[ \operatorname{atan}{\left (\operatorname{asin}{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(asin(x))

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GIAC/XCAS [A]  time = 0.212357, size = 4, normalized size = 1.33 \[ \arctan \left (\arcsin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(arcsin(x))