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

Optimal. Leaf size=18 \[ 2 \tan ^{-1}\left (\sqrt{\frac{1-x}{x+1}}\right ) \]

[Out]

2*ArcTan[Sqrt[(1 - x)/(1 + x)]]

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Rubi [A]  time = 0.0381849, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ 2 \tan ^{-1}\left (\sqrt{\frac{1-x}{x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*ArcTan[Sqrt[(1 - x)/(1 + x)]]

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Rubi in Sympy [A]  time = 2.11338, size = 12, normalized size = 0.67 \[ 2 \operatorname{atan}{\left (\sqrt{\frac{- x + 1}{x + 1}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atan(sqrt((-x + 1)/(x + 1)))

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Mathematica [B]  time = 0.026253, size = 47, normalized size = 2.61 \[ \frac{2 \sqrt{\frac{1-x}{x+1}} \sqrt{1-x^2} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )}{x-1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 30, normalized size = 1.7 \[ -{ \left ( 1+x \right ) \arcsin \left ( x \right ) \sqrt{-{\frac{-1+x}{1+x}}}{\frac{1}{\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.803896, size = 20, normalized size = 1.11 \[ 2 \, \arctan \left (\sqrt{-\frac{x - 1}{x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(sqrt(-(x - 1)/(x + 1)))

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Fricas [A]  time = 0.274667, size = 20, normalized size = 1.11 \[ 2 \, \arctan \left (\sqrt{-\frac{x - 1}{x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(sqrt(-(x - 1)/(x + 1)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \frac{x - 1}{x + 1}}}{x - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-(x - 1)/(x + 1))/(x - 1), x)

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GIAC/XCAS [A]  time = 0.272292, size = 22, normalized size = 1.22 \[ -\frac{1}{2} \, \pi{\rm sign}\left (x + 1\right ) - \arcsin \left (x\right ){\rm sign}\left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*pi*sign(x + 1) - arcsin(x)*sign(x + 1)