3.1110 \(\int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\)

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

[Out]

ArcSin[x]

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Rubi [A]  time = 0.0108689, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcSin[x]

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Rubi in Sympy [A]  time = 2.46221, size = 2, normalized size = 1. \[ \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asin(x)

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Mathematica [A]  time = 0.00722042, size = 2, normalized size = 1. \[ \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcSin[x]

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Maple [B]  time = 0.006, size = 27, normalized size = 13.5 \[{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50029, 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 + 1)*sqrt(-x + 1)),x, algorithm="maxima")

[Out]

arcsin(x)

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Fricas [A]  time = 0.204991, size = 30, normalized size = 15. \[ -2 \, \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.63443, size = 41, normalized size = 20.5 \[ \begin{cases} - 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*I*acosh(sqrt(2)*sqrt(x + 1)/2), Abs(x + 1)/2 > 1), (2*asin(sqrt(2)
*sqrt(x + 1)/2), True))

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GIAC/XCAS [A]  time = 0.205275, size = 18, normalized size = 9. \[ 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arcsin(1/2*sqrt(2)*sqrt(x + 1))