Optimal. Leaf size=35 \[ \frac{1}{2} \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right )-\frac{1}{2} \sqrt{1-\frac{1}{x^2}} x^2 \]
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Rubi [A] time = 0.0466511, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6175, 6178, 266, 47, 63, 206} \[ \frac{1}{2} \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right )-\frac{1}{2} \sqrt{1-\frac{1}{x^2}} x^2 \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 266
Rule 47
Rule 63
Rule 206
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(x)} (1-x) \, dx &=-\int e^{\coth ^{-1}(x)} \left (1-\frac{1}{x}\right ) x \, dx\\ &=\operatorname{Subst}\left (\int \frac{\sqrt{1-x^2}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\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}{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}{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}{2} \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0197955, size = 39, normalized size = 1.11 \[ \frac{1}{2} \log \left (\left (\sqrt{1-\frac{1}{x^2}}+1\right ) x\right )-\frac{1}{2} \sqrt{1-\frac{1}{x^2}} x^2 \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.101, size = 48, normalized size = 1.4 \begin{align*} -{\frac{-1+x}{2} \left ( x\sqrt{{x}^{2}-1}-\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 = 1.0252, size = 112, normalized size = 3.2 \begin{align*} \frac{\left (\frac{x - 1}{x + 1}\right )^{\frac{3}{2}} + \sqrt{\frac{x - 1}{x + 1}}}{\frac{2 \,{\left (x - 1\right )}}{x + 1} - \frac{{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} - 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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Fricas [A] time = 1.82398, size = 151, normalized size = 4.31 \begin{align*} -\frac{1}{2} \,{\left (x^{2} + x\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{x}{\sqrt{\frac{x}{x + 1} - \frac{1}{x + 1}}}\, dx - \int - \frac{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 = 1.14782, size = 149, normalized size = 4.26 \begin{align*} -\frac{\sqrt{\frac{x - 1}{x + 1}} + \frac{1}{\sqrt{\frac{x - 1}{x + 1}}}}{{\left (\sqrt{\frac{x - 1}{x + 1}} + \frac{1}{\sqrt{\frac{x - 1}{x + 1}}}\right )}^{2} - 4} + \frac{1}{4} \, \log \left (\sqrt{\frac{x - 1}{x + 1}} + \frac{1}{\sqrt{\frac{x - 1}{x + 1}}} + 2\right ) - \frac{1}{4} \, \log \left ({\left | \sqrt{\frac{x - 1}{x + 1}} + \frac{1}{\sqrt{\frac{x - 1}{x + 1}}} - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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