Optimal. Leaf size=155 \[ -\frac{3}{2} \log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{3}{2} \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+1\right )-\frac{1}{2} \log \left (\frac{1}{x}+1\right )-\frac{\log (x)}{2}-\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}\right )-\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right ) \]
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Rubi [A] time = 0.0512448, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6171, 105, 60, 91} \[ -\frac{3}{2} \log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{3}{2} \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+1\right )-\frac{1}{2} \log \left (\frac{1}{x}+1\right )-\frac{\log (x)}{2}-\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}\right )-\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Rule 6171
Rule 105
Rule 60
Rule 91
Rubi steps
\begin{align*} \int \frac{e^{\frac{2}{3} \coth ^{-1}(x)}}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x} x} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} (1+x)^{2/3}} \, dx,x,\frac{1}{x}\right )-\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac{1}{x}\right )\\ &=-\sqrt{3} \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} \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{\frac{-1+x}{x}}}{\sqrt{3} \sqrt [3]{1+\frac{1}{x}}}\right )-\frac{3}{2} \log \left (\sqrt [3]{1+\frac{1}{x}}-\sqrt [3]{\frac{-1+x}{x}}\right )-\frac{3}{2} \log \left (1+\frac{\sqrt [3]{\frac{-1+x}{x}}}{\sqrt [3]{1+\frac{1}{x}}}\right )-\frac{1}{2} \log \left (1+\frac{1}{x}\right )-\frac{\log (x)}{2}\\ \end{align*}
Mathematica [C] time = 0.0370847, size = 26, normalized size = 0.17 \[ \frac{3}{2} e^{\frac{8}{3} \coth ^{-1}(x)} \text{Hypergeometric2F1}\left (\frac{2}{3},1,\frac{5}{3},e^{4 \coth ^{-1}(x)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{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 = 1.51635, size = 189, normalized size = 1.22 \begin{align*} -\sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right )}\right ) + \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{1}{2} \, \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{1}{2} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} - \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - \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 = 1.6545, size = 266, normalized size = 1.72 \begin{align*} \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} - 1\right ) + \frac{1}{2} \, \log \left (\frac{{\left (x + 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} +{\left (x - 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + x + 1}{x + 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}{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 = 1.16116, size = 192, normalized size = 1.24 \begin{align*} -\sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right )}\right ) + \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{1}{2} \, \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{1}{2} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} - \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - \log \left ({\left | \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1 \right |}\right ) - \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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