Optimal. Leaf size=22 \[ \tanh ^{-1}\left (\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right ) \]
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Rubi [A] time = 0.0660964, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6175, 6180, 92, 206} \[ \tanh ^{-1}\left (\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right ) \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6180
Rule 92
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(x)}}{1+x} \, dx &=\int \frac{e^{\coth ^{-1}(x)}}{\left (1+\frac{1}{x}\right ) x} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x \sqrt{1+x}} \, dx,x,\frac{1}{x}\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1+\frac{1}{x}} \sqrt{\frac{-1+x}{x}}\right )\\ &=\tanh ^{-1}\left (\sqrt{1+\frac{1}{x}} \sqrt{\frac{-1+x}{x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0094405, size = 18, normalized size = 0.82 \[ \log \left (x \left (\sqrt{\frac{x^2-1}{x^2}}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.126, size = 35, normalized size = 1.6 \begin{align*}{(-1+x)\ln \left ( x+\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}{\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 = 1.00396, size = 42, normalized size = 1.91 \begin{align*} \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt{\frac{x - 1}{x + 1}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57594, size = 88, normalized size = 4. \begin{align*} \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt{\frac{x - 1}{x + 1}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.6437, size = 29, normalized size = 1.32 \begin{align*} - \log{\left (-1 + \frac{1}{\sqrt{1 - \frac{2}{x + 1}}} \right )} + \log{\left (1 + \frac{1}{\sqrt{1 - \frac{2}{x + 1}}} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14995, size = 43, normalized size = 1.95 \begin{align*} \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \log \left ({\left | \sqrt{\frac{x - 1}{x + 1}} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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