Optimal. Leaf size=47 \[ \frac{2 \left (\frac{1}{x}+1\right )}{\sqrt{1-\frac{1}{x^2}}}-\sqrt{1-\frac{1}{x^2}} x-2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right ) \]
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Rubi [A] time = 0.135316, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {6175, 6177, 852, 1805, 807, 266, 63, 206} \[ \frac{2 \left (\frac{1}{x}+1\right )}{\sqrt{1-\frac{1}{x^2}}}-\sqrt{1-\frac{1}{x^2}} x-2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6177
Rule 852
Rule 1805
Rule 807
Rule 266
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(x)} x}{1-x} \, dx &=-\int \frac{e^{\coth ^{-1}(x)}}{1-\frac{1}{x}} \, dx\\ &=\operatorname{Subst}\left (\int \frac{\sqrt{1-x^2}}{(1-x)^2 x^2} \, dx,x,\frac{1}{x}\right )\\ &=\operatorname{Subst}\left (\int \frac{(1+x)^2}{x^2 \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{-1-2 x}{x^2 \sqrt{1-x^2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 \left (1+\frac{1}{x}\right )}{\sqrt{1-\frac{1}{x^2}}}-\sqrt{1-\frac{1}{x^2}} 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}}}-\sqrt{1-\frac{1}{x^2}} x+\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}}}-\sqrt{1-\frac{1}{x^2}} 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}}}-\sqrt{1-\frac{1}{x^2}} x-2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0521181, size = 41, normalized size = 0.87 \[ -\frac{\sqrt{1-\frac{1}{x^2}} (x-3) x}{x-1}-2 \log \left (\left (\sqrt{1-\frac{1}{x^2}}+1\right ) x\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.118, size = 106, normalized size = 2.3 \begin{align*}{\frac{1}{-1+x} \left ( \left ({x}^{2}-1 \right ) ^{{\frac{3}{2}}}-2\,{x}^{2}\sqrt{{x}^{2}-1}-2\,\ln \left ( x+\sqrt{{x}^{2}-1} \right ){x}^{2}+4\,x\sqrt{{x}^{2}-1}+4\,\ln \left ( x+\sqrt{{x}^{2}-1} \right ) x-2\,\sqrt{{x}^{2}-1}-2\,\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 = 1.03955, size = 100, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (\frac{2 \,{\left (x - 1\right )}}{x + 1} - 1\right )}}{\left (\frac{x - 1}{x + 1}\right )^{\frac{3}{2}} - \sqrt{\frac{x - 1}{x + 1}}} - 2 \, \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) + 2 \, \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 = 1.67737, size = 184, normalized size = 3.91 \begin{align*} -\frac{2 \,{\left (x - 1\right )} \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - 2 \,{\left (x - 1\right )} \log \left (\sqrt{\frac{x - 1}{x + 1}} - 1\right ) +{\left (x^{2} - 2 \, x - 3\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{x}{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 [B] time = 1.15349, size = 113, normalized size = 2.4 \begin{align*} \frac{2 \,{\left (\frac{2 \,{\left (x - 1\right )}}{x + 1} - 1\right )}}{\frac{{\left (x - 1\right )} \sqrt{\frac{x - 1}{x + 1}}}{x + 1} - \sqrt{\frac{x - 1}{x + 1}}} - 2 \, \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) + 2 \, \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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