Optimal. Leaf size=96 \[ \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x-\log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{\log (x)}{3}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0286798, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6170, 94, 91} \[ \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x-\log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{\log (x)}{3}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 6170
Rule 94
Rule 91
Rubi steps
\begin{align*} \int e^{\frac{2}{3} \coth ^{-1}(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^2} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt [3]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{2/3} x-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt [3]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{2/3} x-\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 )}{\sqrt{3}}-\log \left (\sqrt [3]{1+\frac{1}{x}}-\sqrt [3]{-\frac{1-x}{x}}\right )-\frac{\log (x)}{3}\\ \end{align*}
Mathematica [A] time = 0.160126, size = 85, normalized size = 0.89 \[ \frac{1}{3} \left (\frac{6 e^{\frac{2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}-1}-2 \log \left (1-e^{\frac{2}{3} \coth ^{-1}(x)}\right )+\log \left (e^{\frac{2}{3} \coth ^{-1}(x)}+e^{\frac{4}{3} \coth ^{-1}(x)}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 e^{\frac{2}{3} \coth ^{-1}(x)}+1}{\sqrt{3}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\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 = 1.54199, size = 130, normalized size = 1.35 \begin{align*} -\frac{2}{3} \, \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{x - 1}{x + 1}\right )^{\frac{2}{3}}}{\frac{x - 1}{x + 1} - 1} + \frac{1}{3} \, \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}{3} \, \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 = 1.5966, size = 282, normalized size = 2.94 \begin{align*}{\left (x + 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} - \frac{2}{3} \, \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}{3} \, \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}{3} \, \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{1}{\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 = 1.1857, size = 131, normalized size = 1.36 \begin{align*} -\frac{2}{3} \, \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{x - 1}{x + 1}\right )^{\frac{2}{3}}}{\frac{x - 1}{x + 1} - 1} + \frac{1}{3} \, \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}{3} \, \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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