Optimal. Leaf size=99 \[ \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3}-\log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+1\right )-\frac{1}{3} \log \left (\frac{1}{x}+1\right )-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0429347, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6171, 50, 60} \[ \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3}-\log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+1\right )-\frac{1}{3} \log \left (\frac{1}{x}+1\right )-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 50
Rule 60
Rubi steps
\begin{align*} \int \frac{e^{\frac{2}{3} \coth ^{-1}(x)}}{x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt [3]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{2/3}-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-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}-\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 (1+\frac{\sqrt [3]{-\frac{1-x}{x}}}{\sqrt [3]{1+\frac{1}{x}}}\right )-\frac{1}{3} \log \left (1+\frac{1}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.144131, size = 87, normalized size = 0.88 \[ \frac{2 e^{\frac{2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}+1}-\frac{2}{3} \log \left (e^{\frac{2}{3} \coth ^{-1}(x)}+1\right )+\frac{1}{3} \log \left (-e^{\frac{2}{3} \coth ^{-1}(x)}+e^{\frac{4}{3} \coth ^{-1}(x)}+1\right )-\frac{2 \tan ^{-1}\left (\frac{2 e^{\frac{2}{3} \coth ^{-1}(x)}-1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.085, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}{\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 = 1.62854, size = 132, normalized size = 1.33 \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 = 1.64112, size = 293, normalized size = 2.96 \begin{align*} \frac{2 \, \sqrt{3} x \arctan \left (\frac{2}{3} \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + x \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} - \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - 2 \, x \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) + 3 \,{\left (x + 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}}}{3 \, x} \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^{2} \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 = 1.13534, size = 134, 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 | \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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