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