Optimal. Leaf size=41 \[ 2 \tan ^{-1}\left (\sqrt{\frac{x+1}{1-x}}\right )-(1-x) \sqrt{\frac{x+1}{1-x}} \]
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Rubi [A] time = 0.0130075, 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, Rules used = {1959, 288, 203} \[ 2 \tan ^{-1}\left (\sqrt{\frac{x+1}{1-x}}\right )-(1-x) \sqrt{\frac{x+1}{1-x}} \]
Antiderivative was successfully verified.
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Rule 1959
Rule 288
Rule 203
Rubi steps
\begin{align*} \int \sqrt{\frac{1+x}{1-x}} \, dx &=4 \operatorname{Subst}\left (\int \frac{x^2}{\left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1+x}{1-x}}\right )\\ &=-(1-x) \sqrt{\frac{1+x}{1-x}}+2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\frac{1+x}{1-x}}\right )\\ &=-(1-x) \sqrt{\frac{1+x}{1-x}}+2 \tan ^{-1}\left (\sqrt{\frac{1+x}{1-x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0228433, size = 62, normalized size = 1.51 \[ \frac{\sqrt{\frac{x+1}{1-x}} \left ((x-1) \sqrt{x+1}-2 \sqrt{1-x} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )\right )}{\sqrt{x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 41, normalized size = 1. \begin{align*}{(-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 ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41076, size = 58, normalized size = 1.41 \begin{align*} \frac{2 \, \sqrt{-\frac{x + 1}{x - 1}}}{\frac{x + 1}{x - 1} - 1} + 2 \, \arctan \left (\sqrt{-\frac{x + 1}{x - 1}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27597, size = 90, normalized size = 2.2 \begin{align*}{\left (x - 1\right )} \sqrt{-\frac{x + 1}{x - 1}} + 2 \, \arctan \left (\sqrt{-\frac{x + 1}{x - 1}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{x + 1}{1 - x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07219, size = 41, normalized size = 1. \begin{align*} \frac{1}{2} \, \pi \mathrm{sgn}\left (x - 1\right ) - \arcsin \left (x\right ) \mathrm{sgn}\left (x - 1\right ) + \sqrt{-x^{2} + 1} \mathrm{sgn}\left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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