Optimal. Leaf size=24 \[ -\frac{\left (1-\frac{1}{x^2}\right )^{3/2}}{3 \left (1-\frac{1}{x}\right )^3} \]
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Rubi [A] time = 0.0714459, antiderivative size = 24, 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 = {6175, 6178, 651} \[ -\frac{\left (1-\frac{1}{x^2}\right )^{3/2}}{3 \left (1-\frac{1}{x}\right )^3} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx &=\int \frac{e^{\coth ^{-1}(x)}}{\left (1-\frac{1}{x}\right )^2 x^2} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{\sqrt{1-x^2}}{(1-x)^3} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\left (1-\frac{1}{x^2}\right )^{3/2}}{3 \left (1-\frac{1}{x}\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0143856, size = 24, normalized size = 1. \[ -\frac{\sqrt{1-\frac{1}{x^2}} x (x+1)}{3 (x-1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 22, normalized size = 0.9 \begin{align*} -{\frac{1+x}{-3+3\,x}{\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 = 1.06845, size = 18, normalized size = 0.75 \begin{align*} -\frac{1}{3 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5881, size = 81, normalized size = 3.38 \begin{align*} -\frac{{\left (x^{2} + 2 \, x + 1\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{1}{\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 = 1.14127, size = 28, normalized size = 1.17 \begin{align*} -\frac{x + 1}{3 \,{\left (x - 1\right )} \sqrt{\frac{x - 1}{x + 1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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