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

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

[Out]

ArcSin[x]

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Rubi [A]  time = 0.00441641, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 - x^2],x]

[Out]

ArcSin[x]

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Rubi in Sympy [A]  time = 0.098879, size = 2, normalized size = 1. \[ \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asin(x)

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Mathematica [A]  time = 0.00629055, size = 2, normalized size = 1. \[ \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[1 - x^2],x]

[Out]

ArcSin[x]

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Maple [A]  time = 0.003, size = 3, normalized size = 1.5 \[ \arcsin \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^2+1)^(1/2),x)

[Out]

arcsin(x)

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Maxima [A]  time = 0.755059, size = 3, normalized size = 1.5 \[ \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsin(x)

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Fricas [A]  time = 0.267103, size = 24, normalized size = 12. \[ -2 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*arctan((sqrt(-x^2 + 1) - 1)/x)

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Sympy [A]  time = 0.29004, size = 2, normalized size = 1. \[ \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**2+1)**(1/2),x)

[Out]

asin(x)

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GIAC/XCAS [A]  time = 0.266818, size = 3, normalized size = 1.5 \[ \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsin(x)