Optimal. Leaf size=23 \[ -\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.028107, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6129, 63, 206} \[ -\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 6129
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(x)}}{(1+x)^{3/2}} \, dx &=\int \frac{1}{\sqrt{1-x} (1+x)} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1-x}\right )\right )\\ &=-\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0043876, size = 23, normalized size = 1. \[ -\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 40, normalized size = 1.7 \begin{align*} -{\sqrt{2}\sqrt{-{x}^{2}+1}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{1-x}} \right ){\frac{1}{\sqrt{1-x}}}{\frac{1}{\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{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 = 1.78585, size = 122, normalized size = 5.3 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{x^{2} + 2 \, \sqrt{2} \sqrt{-x^{2} + 1} \sqrt{x + 1} - 2 \, x - 3}{x^{2} + 2 \, 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 \frac{1}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \sqrt{x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20302, size = 50, normalized size = 2.17 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (\sqrt{2} + \sqrt{-x + 1}\right ) + \frac{1}{2} \, \sqrt{2} \log \left (\sqrt{2} - \sqrt{-x + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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