Optimal. Leaf size=130 \[ \frac{1}{2} \left (\frac{1}{x}+1\right )^{4/3} \left (\frac{x-1}{x}\right )^{2/3} x^2+\frac{1}{3} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x-\frac{1}{3} \log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{\log (x)}{9}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0466163, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6171, 96, 94, 91} \[ \frac{1}{2} \left (\frac{1}{x}+1\right )^{4/3} \left (\frac{x-1}{x}\right )^{2/3} x^2+\frac{1}{3} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x-\frac{1}{3} \log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{\log (x)}{9}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 96
Rule 94
Rule 91
Rubi steps
\begin{align*} \int e^{\frac{2}{3} \coth ^{-1}(x)} x \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} \left (1+\frac{1}{x}\right )^{4/3} \left (\frac{-1+x}{x}\right )^{2/3} x^2-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3} x+\frac{1}{2} \left (1+\frac{1}{x}\right )^{4/3} \left (\frac{-1+x}{x}\right )^{2/3} x^2-\frac{2}{9} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \sqrt [3]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{2/3} x+\frac{1}{2} \left (1+\frac{1}{x}\right )^{4/3} \left (\frac{-1+x}{x}\right )^{2/3} x^2-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{-\frac{1-x}{x}}}{\sqrt{3} \sqrt [3]{1+\frac{1}{x}}}\right )}{3 \sqrt{3}}-\frac{1}{3} \log \left (\sqrt [3]{1+\frac{1}{x}}-\sqrt [3]{-\frac{1-x}{x}}\right )-\frac{\log (x)}{9}\\ \end{align*}
Mathematica [A] time = 0.429379, size = 165, normalized size = 1.27 \[ \frac{1}{9} \left (\frac{24 e^{\frac{2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}-1}+\frac{18 e^{\frac{2}{3} \coth ^{-1}(x)}}{\left (e^{2 \coth ^{-1}(x)}-1\right )^2}-2 \log \left (1-e^{\frac{1}{3} \coth ^{-1}(x)}\right )-2 \log \left (e^{\frac{1}{3} \coth ^{-1}(x)}+1\right )+\log \left (-e^{\frac{1}{3} \coth ^{-1}(x)}+e^{\frac{2}{3} \coth ^{-1}(x)}+1\right )+\log \left (e^{\frac{1}{3} \coth ^{-1}(x)}+e^{\frac{2}{3} \coth ^{-1}(x)}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 e^{\frac{1}{3} \coth ^{-1}(x)}-1}{\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 e^{\frac{1}{3} \coth ^{-1}(x)}+1}{\sqrt{3}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{x{\frac{1}{\sqrt [3]{{\frac{-1+x}{1+x}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51746, size = 166, normalized size = 1.28 \begin{align*} -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right )}\right ) + \frac{2 \,{\left (\left (\frac{x - 1}{x + 1}\right )^{\frac{5}{3}} - 4 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}\right )}}{3 \,{\left (\frac{2 \,{\left (x - 1\right )}}{x + 1} - \frac{{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1\right )}} + \frac{1}{9} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - \frac{2}{9} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56023, size = 301, normalized size = 2.32 \begin{align*} \frac{1}{6} \,{\left (3 \, x^{2} + 8 \, x + 5\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} - \frac{2}{9} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + \frac{1}{9} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - \frac{2}{9} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 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 [3]{\frac{x - 1}{x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1516, size = 162, normalized size = 1.25 \begin{align*} -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right )}\right ) - \frac{2 \,{\left (\frac{{\left (x - 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}}{x + 1} - 4 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}\right )}}{3 \,{\left (\frac{x - 1}{x + 1} - 1\right )}^{2}} + \frac{1}{9} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - \frac{2}{9} \, \log \left ({\left | \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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