Optimal. Leaf size=18 \[ -\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{3/2} x^3 \]
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Rubi [A] time = 0.0525726, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6175, 6178, 264} \[ -\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{3/2} x^3 \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 264
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(x)} (1-x) x \, dx &=-\int e^{\coth ^{-1}(x)} \left (1-\frac{1}{x}\right ) x^2 \, dx\\ &=\operatorname{Subst}\left (\int \frac{\sqrt{1-x^2}}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{3/2} x^3\\ \end{align*}
Mathematica [A] time = 0.0199014, size = 21, normalized size = 1.17 \[ -\frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \left (x^2-1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 22, normalized size = 1.2 \begin{align*} -{\frac{ \left ( 1+x \right ) \left ( -1+x \right ) ^{2}}{3}{\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 [B] time = 1.01459, size = 68, normalized size = 3.78 \begin{align*} \frac{8 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{3}{2}}}{3 \,{\left (\frac{3 \,{\left (x - 1\right )}}{x + 1} - \frac{3 \,{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac{{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83827, size = 65, normalized size = 3.61 \begin{align*} -\frac{1}{3} \,{\left (x^{3} + x^{2} - 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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{x}{\sqrt{\frac{x}{x + 1} - \frac{1}{x + 1}}}\, dx - \int \frac{x^{2}}{\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 = 1.16763, size = 39, normalized size = 2.17 \begin{align*} \frac{8}{3 \,{\left (\sqrt{\frac{x - 1}{x + 1}} - \frac{1}{\sqrt{\frac{x - 1}{x + 1}}}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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