Optimal. Leaf size=31 \[ 2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-2 \sqrt{x+1} \]
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Rubi [A] time = 0.0336543, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {6127, 627, 50, 63, 206} \[ 2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-2 \sqrt{x+1} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 627
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(x)}}{\sqrt{1-x}} \, dx &=\int \frac{\sqrt{1-x^2}}{(1-x)^{3/2}} \, dx\\ &=\int \frac{\sqrt{1+x}}{1-x} \, dx\\ &=-2 \sqrt{1+x}+2 \int \frac{1}{(1-x) \sqrt{1+x}} \, dx\\ &=-2 \sqrt{1+x}+4 \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+x}\right )\\ &=-2 \sqrt{1+x}+2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1+x}}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0054061, size = 31, normalized size = 1. \[ 2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-2 \sqrt{x+1} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.037, size = 52, normalized size = 1.7 \begin{align*} -2\,{\frac{\sqrt{-{x}^{2}+1}\sqrt{1-x} \left ({\it Artanh} \left ( 1/2\,\sqrt{1+x}\sqrt{2} \right ) \sqrt{2}-\sqrt{1+x} \right ) }{ \left ( -1+x \right ) \sqrt{1+x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{\sqrt{-x^{2} + 1} \sqrt{-x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82272, size = 185, normalized size = 5.97 \begin{align*} \frac{\sqrt{2}{\left (x - 1\right )} \log \left (-\frac{x^{2} - 2 \, \sqrt{2} \sqrt{-x^{2} + 1} \sqrt{-x + 1} + 2 \, x - 3}{x^{2} - 2 \, x + 1}\right ) + 2 \, \sqrt{-x^{2} + 1} \sqrt{-x + 1}}{x - 1} \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 \frac{x + 1}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \sqrt{1 - x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20945, size = 50, normalized size = 1.61 \begin{align*} -\sqrt{2} \log \left (\frac{\sqrt{2} - \sqrt{x + 1}}{\sqrt{2} + \sqrt{x + 1}}\right ) - 2 \, \sqrt{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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