Optimal. Leaf size=41 \[ 2 \tan ^{-1}\left (\sqrt{\frac{x+1}{1-x}}\right )-(1-x) \sqrt{\frac{x+1}{1-x}} \]
[Out]
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Rubi [A] time = 0.0304611, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 2 \tan ^{-1}\left (\sqrt{\frac{x+1}{1-x}}\right )-(1-x) \sqrt{\frac{x+1}{1-x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[(1 + x)/(1 - x)],x]
[Out]
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Rubi in Sympy [A] time = 1.69038, size = 32, normalized size = 0.78 \[ - \frac{2 \sqrt{\frac{x + 1}{- x + 1}}}{1 + \frac{x + 1}{- x + 1}} + 2 \operatorname{atan}{\left (\sqrt{\frac{x + 1}{- x + 1}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((1+x)/(1-x))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0419962, size = 60, normalized size = 1.46 \[ \frac{\sqrt{\frac{x+1}{1-x}} \left (\sqrt{x+1} (x-1)+2 \sqrt{1-x} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )\right )}{\sqrt{x+1}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[(1 + x)/(1 - x)],x]
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Maple [A] time = 0.009, size = 41, normalized size = 1. \[{(-1+x)\sqrt{-{\frac{1+x}{-1+x}}} \left ( \sqrt{-{x}^{2}+1}-\arcsin \left ( x \right ) \right ){\frac{1}{\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((1+x)/(1-x))^(1/2),x)
[Out]
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Maxima [A] time = 1.53034, size = 58, normalized size = 1.41 \[ \frac{2 \, \sqrt{-\frac{x + 1}{x - 1}}}{\frac{x + 1}{x - 1} - 1} + 2 \, \arctan \left (\sqrt{-\frac{x + 1}{x - 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-(x + 1)/(x - 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.208365, size = 43, normalized size = 1.05 \[{\left (x - 1\right )} \sqrt{-\frac{x + 1}{x - 1}} + 2 \, \arctan \left (\sqrt{-\frac{x + 1}{x - 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-(x + 1)/(x - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x + 1}{- x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((1+x)/(1-x))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210591, size = 41, normalized size = 1. \[ \frac{1}{2} \, \pi{\rm sign}\left (x - 1\right ) - \arcsin \left (x\right ){\rm sign}\left (x - 1\right ) + \sqrt{-x^{2} + 1}{\rm sign}\left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-(x + 1)/(x - 1)),x, algorithm="giac")
[Out]