Optimal. Leaf size=53 \[ \frac{1}{3} \left (1-\frac{1}{x^2}\right )^{3/2} x^3-\frac{1}{2} \sqrt{1-\frac{1}{x^2}} x^2+\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right ) \]
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Rubi [A] time = 0.0904612, antiderivative size = 53, 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, 807, 266, 47, 63, 206} \[ \frac{1}{3} \left (1-\frac{1}{x^2}\right )^{3/2} x^3-\frac{1}{2} \sqrt{1-\frac{1}{x^2}} x^2+\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 807
Rule 266
Rule 47
Rule 63
Rule 206
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(x)} (1-x)^2 \, dx &=\int e^{\coth ^{-1}(x)} \left (1-\frac{1}{x}\right )^2 x^2 \, dx\\ &=-\operatorname{Subst}\left (\int \frac{(1-x) \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+\operatorname{Subst}\left (\int \frac{\sqrt{1-x^2}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{3/2} x^3+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{1-x}}{x^2} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{2} \sqrt{1-\frac{1}{x^2}} x^2+\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{3/2} x^3-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{2} \sqrt{1-\frac{1}{x^2}} x^2+\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{3/2} x^3+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-\frac{1}{x^2}}\right )\\ &=-\frac{1}{2} \sqrt{1-\frac{1}{x^2}} x^2+\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{3/2} x^3+\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0318873, size = 47, normalized size = 0.89 \[ \frac{1}{6} \sqrt{1-\frac{1}{x^2}} x \left (2 x^2-3 x-2\right )+\frac{1}{2} \log \left (\left (\sqrt{1-\frac{1}{x^2}}+1\right ) x\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.113, size = 60, normalized size = 1.1 \begin{align*}{\frac{-1+x}{6} \left ( 2\, \left ( \left ( 1+x \right ) \left ( -1+x \right ) \right ) ^{3/2}-3\,x\sqrt{{x}^{2}-1}+3\,\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \right ){\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}{\frac{1}{\sqrt{ \left ( 1+x \right ) \left ( -1+x \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12716, size = 151, normalized size = 2.85 \begin{align*} -\frac{3 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{2}} + 8 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{3}{2}} - 3 \, \sqrt{\frac{x - 1}{x + 1}}}{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 )}} + \frac{1}{2} \, \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \frac{1}{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 = 1.57553, size = 169, normalized size = 3.19 \begin{align*} \frac{1}{6} \,{\left (2 \, x^{3} - x^{2} - 5 \, x - 2\right )} \sqrt{\frac{x - 1}{x + 1}} + \frac{1}{2} \, \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \frac{1}{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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x - 1\right )^{2}}{\sqrt{\frac{x - 1}{x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13403, size = 144, normalized size = 2.72 \begin{align*} -\frac{\frac{8 \,{\left (x - 1\right )} \sqrt{\frac{x - 1}{x + 1}}}{x + 1} + \frac{3 \,{\left (x - 1\right )}^{2} \sqrt{\frac{x - 1}{x + 1}}}{{\left (x + 1\right )}^{2}} - 3 \, \sqrt{\frac{x - 1}{x + 1}}}{3 \,{\left (\frac{x - 1}{x + 1} - 1\right )}^{3}} + \frac{1}{2} \, \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \frac{1}{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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