Optimal. Leaf size=33 \[ \frac{2 \left (\frac{1}{x}+1\right )}{\sqrt{1-\frac{1}{x^2}}}-\tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right ) \]
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Rubi [A] time = 0.131617, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6175, 6178, 852, 1805, 266, 63, 206} \[ \frac{2 \left (\frac{1}{x}+1\right )}{\sqrt{1-\frac{1}{x^2}}}-\tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 852
Rule 1805
Rule 266
Rule 63
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{\sqrt{1-x^2}}{(1-x)^2 x} \, dx,x,\frac{1}{x}\right )\\ &=\operatorname{Subst}\left (\int \frac{(1+x)^2}{x \left (1-x^2\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 \left (1+\frac{1}{x}\right )}{\sqrt{1-\frac{1}{x^2}}}+\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-x^2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 \left (1+\frac{1}{x}\right )}{\sqrt{1-\frac{1}{x^2}}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{2 \left (1+\frac{1}{x}\right )}{\sqrt{1-\frac{1}{x^2}}}-\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-\frac{1}{x^2}}\right )\\ &=\frac{2 \left (1+\frac{1}{x}\right )}{\sqrt{1-\frac{1}{x^2}}}-\tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0227887, size = 38, normalized size = 1.15 \[ \frac{2 \sqrt{1-\frac{1}{x^2}} x}{x-1}-\log \left (\left (\sqrt{1-\frac{1}{x^2}}+1\right ) x\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.12, size = 106, normalized size = 3.2 \begin{align*}{\frac{1}{-1+x} \left ( \left ({x}^{2}-1 \right ) ^{{\frac{3}{2}}}-{x}^{2}\sqrt{{x}^{2}-1}-\ln \left ( x+\sqrt{{x}^{2}-1} \right ){x}^{2}+2\,x\sqrt{{x}^{2}-1}+2\,\ln \left ( x+\sqrt{{x}^{2}-1} \right ) x-\sqrt{{x}^{2}-1}-\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \right ){\frac{1}{\sqrt{ \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00447, size = 59, normalized size = 1.79 \begin{align*} \frac{2}{\sqrt{\frac{x - 1}{x + 1}}} - \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 [B] time = 1.58125, size = 170, normalized size = 5.15 \begin{align*} -\frac{{\left (x - 1\right )} \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) -{\left (x - 1\right )} \log \left (\sqrt{\frac{x - 1}{x + 1}} - 1\right ) - 2 \,{\left (x + 1\right )} \sqrt{\frac{x - 1}{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{1}{x \sqrt{\frac{x}{x + 1} - \frac{1}{x + 1}} - \sqrt{\frac{x}{x + 1} - \frac{1}{x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16523, size = 61, normalized size = 1.85 \begin{align*} \frac{2}{\sqrt{\frac{x - 1}{x + 1}}} - \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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