Optimal. Leaf size=55 \[ -\frac{4 \left (\frac{1}{x}+1\right )}{3 \left (1-\frac{1}{x^2}\right )^{3/2}}-\frac{\frac{5}{x}+3}{3 \sqrt{1-\frac{1}{x^2}}}+\tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right ) \]
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Rubi [A] time = 0.15491, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {6175, 6178, 852, 1805, 823, 12, 266, 63, 206} \[ -\frac{4 \left (\frac{1}{x}+1\right )}{3 \left (1-\frac{1}{x^2}\right )^{3/2}}-\frac{\frac{5}{x}+3}{3 \sqrt{1-\frac{1}{x^2}}}+\tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 852
Rule 1805
Rule 823
Rule 12
Rule 266
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(x)} x}{(1-x)^2} \, dx &=\int \frac{e^{\coth ^{-1}(x)}}{\left (1-\frac{1}{x}\right )^2 x} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{\sqrt{1-x^2}}{(1-x)^3 x} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \frac{(1+x)^3}{x \left (1-x^2\right )^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 \left (1+\frac{1}{x}\right )}{3 \left (1-\frac{1}{x^2}\right )^{3/2}}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{-3-5 x}{x \left (1-x^2\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 \left (1+\frac{1}{x}\right )}{3 \left (1-\frac{1}{x^2}\right )^{3/2}}-\frac{3+\frac{5}{x}}{3 \sqrt{1-\frac{1}{x^2}}}+\frac{1}{3} \operatorname{Subst}\left (\int -\frac{3}{x \sqrt{1-x^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 \left (1+\frac{1}{x}\right )}{3 \left (1-\frac{1}{x^2}\right )^{3/2}}-\frac{3+\frac{5}{x}}{3 \sqrt{1-\frac{1}{x^2}}}-\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-x^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 \left (1+\frac{1}{x}\right )}{3 \left (1-\frac{1}{x^2}\right )^{3/2}}-\frac{3+\frac{5}{x}}{3 \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{4 \left (1+\frac{1}{x}\right )}{3 \left (1-\frac{1}{x^2}\right )^{3/2}}-\frac{3+\frac{5}{x}}{3 \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{4 \left (1+\frac{1}{x}\right )}{3 \left (1-\frac{1}{x^2}\right )^{3/2}}-\frac{3+\frac{5}{x}}{3 \sqrt{1-\frac{1}{x^2}}}+\tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0553172, size = 43, normalized size = 0.78 \[ \frac{\sqrt{1-\frac{1}{x^2}} (5-7 x) x}{3 (x-1)^2}+\log \left (\left (\sqrt{1-\frac{1}{x^2}}+1\right ) x\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.116, size = 146, normalized size = 2.7 \begin{align*} -{\frac{1}{3\, \left ( -1+x \right ) ^{2}} \left ( 3\,x \left ({x}^{2}-1 \right ) ^{3/2}-3\,{x}^{3}\sqrt{{x}^{2}-1}-3\,\ln \left ( x+\sqrt{{x}^{2}-1} \right ){x}^{3}-2\, \left ({x}^{2}-1 \right ) ^{3/2}+9\,{x}^{2}\sqrt{{x}^{2}-1}+9\,\ln \left ( x+\sqrt{{x}^{2}-1} \right ){x}^{2}-9\,x\sqrt{{x}^{2}-1}-9\,\ln \left ( x+\sqrt{{x}^{2}-1} \right ) x+3\,\sqrt{{x}^{2}-1}+3\,\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 = 1.00905, size = 76, normalized size = 1.38 \begin{align*} -\frac{\frac{6 \,{\left (x - 1\right )}}{x + 1} + 1}{3 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{3}{2}}} + \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 = 1.57667, size = 223, normalized size = 4.05 \begin{align*} \frac{3 \,{\left (x^{2} - 2 \, x + 1\right )} \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - 3 \,{\left (x^{2} - 2 \, x + 1\right )} \log \left (\sqrt{\frac{x - 1}{x + 1}} - 1\right ) -{\left (7 \, x^{2} + 2 \, x - 5\right )} \sqrt{\frac{x - 1}{x + 1}}}{3 \,{\left (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{x}{\sqrt{\frac{x - 1}{x + 1}} \left (x - 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14982, size = 88, normalized size = 1.6 \begin{align*} -\frac{{\left (x + 1\right )}{\left (\frac{6 \,{\left (x - 1\right )}}{x + 1} + 1\right )}}{3 \,{\left (x - 1\right )} \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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