Optimal. Leaf size=402 \[ -\frac{1}{2} \log \left (\frac{\sqrt [3]{\frac{1}{x}+1}}{\sqrt [3]{\frac{x-1}{x}}}-\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1\right )+\frac{1}{2} \log \left (\frac{\sqrt [3]{\frac{1}{x}+1}}{\sqrt [3]{\frac{x-1}{x}}}+\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1\right )+\frac{1}{2} \sqrt{3} \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}-\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )-\frac{1}{2} \sqrt{3} \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )-\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}}{\sqrt{3}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1}{\sqrt{3}}\right )-\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right )+\tan ^{-1}\left (\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+\sqrt{3}\right )+2 \tan ^{-1}\left (\frac{\sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}\right ) \]
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Rubi [A] time = 0.525113, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 13, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.083, Rules used = {6171, 105, 63, 331, 295, 634, 618, 204, 628, 203, 93, 210, 206} \[ -\frac{1}{2} \log \left (\frac{\sqrt [3]{\frac{1}{x}+1}}{\sqrt [3]{\frac{x-1}{x}}}-\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1\right )+\frac{1}{2} \log \left (\frac{\sqrt [3]{\frac{1}{x}+1}}{\sqrt [3]{\frac{x-1}{x}}}+\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1\right )+\frac{1}{2} \sqrt{3} \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}-\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )-\frac{1}{2} \sqrt{3} \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )-\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}}{\sqrt{3}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1}{\sqrt{3}}\right )-\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right )+\tan ^{-1}\left (\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+\sqrt{3}\right )+2 \tan ^{-1}\left (\frac{\sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}\right ) \]
Antiderivative was successfully verified.
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Rule 6171
Rule 105
Rule 63
Rule 331
Rule 295
Rule 634
Rule 618
Rule 204
Rule 628
Rule 203
Rule 93
Rule 210
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{3} \coth ^{-1}(x)}}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x} x} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx,x,\frac{1}{x}\right )-\operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx,x,\frac{1}{x}\right )\\ &=6 \operatorname{Subst}\left (\int \frac{x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{\frac{-1+x}{x}}\right )-6 \operatorname{Subst}\left (\int \frac{1}{-1+x^6} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+2 \operatorname{Subst}\left (\int \frac{1-\frac{x}{2}}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+2 \operatorname{Subst}\left (\int \frac{1+\frac{x}{2}}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+6 \operatorname{Subst}\left (\int \frac{x^4}{1+x^6} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )\\ &=2 \tanh ^{-1}\left (\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1+2 x}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+2 \operatorname{Subst}\left (\int \frac{-\frac{1}{2}+\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+2 \operatorname{Subst}\left (\int \frac{-\frac{1}{2}-\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )\\ &=2 \tan ^{-1}\left (\frac{\sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )-\frac{1}{2} \log \left (1+\frac{\sqrt [3]{1+\frac{1}{x}}}{\sqrt [3]{-\frac{1-x}{x}}}-\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )+\frac{1}{2} \log \left (1+\frac{\sqrt [3]{1+\frac{1}{x}}}{\sqrt [3]{-\frac{1-x}{x}}}+\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )-3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+\frac{2 \sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )-3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+\frac{1}{2} \sqrt{3} \operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )-\frac{1}{2} \sqrt{3} \operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )\\ &=-\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}}{\sqrt{3}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}}{\sqrt{3}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )-\frac{1}{2} \log \left (1+\frac{\sqrt [3]{1+\frac{1}{x}}}{\sqrt [3]{-\frac{1-x}{x}}}-\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )+\frac{1}{2} \log \left (1+\frac{\sqrt [3]{1+\frac{1}{x}}}{\sqrt [3]{-\frac{1-x}{x}}}+\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )+\frac{1}{2} \sqrt{3} \log \left (1-\frac{\sqrt{3} \sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}+\frac{\sqrt [3]{-\frac{1-x}{x}}}{\sqrt [3]{1+\frac{1}{x}}}\right )-\frac{1}{2} \sqrt{3} \log \left (1+\frac{\sqrt{3} \sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}+\frac{\sqrt [3]{-\frac{1-x}{x}}}{\sqrt [3]{1+\frac{1}{x}}}\right )-\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+\frac{2 \sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )-\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+\frac{2 \sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )\\ &=-\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}}{\sqrt{3}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}}{\sqrt{3}}\right )-\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )-\frac{1}{2} \log \left (1+\frac{\sqrt [3]{1+\frac{1}{x}}}{\sqrt [3]{-\frac{1-x}{x}}}-\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )+\frac{1}{2} \log \left (1+\frac{\sqrt [3]{1+\frac{1}{x}}}{\sqrt [3]{-\frac{1-x}{x}}}+\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )+\frac{1}{2} \sqrt{3} \log \left (1-\frac{\sqrt{3} \sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}+\frac{\sqrt [3]{-\frac{1-x}{x}}}{\sqrt [3]{1+\frac{1}{x}}}\right )-\frac{1}{2} \sqrt{3} \log \left (1+\frac{\sqrt{3} \sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}+\frac{\sqrt [3]{-\frac{1-x}{x}}}{\sqrt [3]{1+\frac{1}{x}}}\right )\\ \end{align*}
Mathematica [C] time = 0.0317729, size = 26, normalized size = 0.06 \[ \frac{12}{7} e^{\frac{7}{3} \coth ^{-1}(x)} \text{Hypergeometric2F1}\left (\frac{7}{12},1,\frac{19}{12},e^{4 \coth ^{-1}(x)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.068, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt [6]{{\frac{-1+x}{1+x}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83976, size = 1080, normalized size = 2.69 \begin{align*} -\sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + \frac{1}{3} \, \sqrt{3}\right ) - \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} - \frac{1}{3} \, \sqrt{3}\right ) - \frac{1}{2} \, \sqrt{3} \log \left (16 \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 16 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 16\right ) + \frac{1}{2} \, \sqrt{3} \log \left (-16 \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 16 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 16\right ) - 2 \, \arctan \left (\sqrt{3} + \frac{1}{2} \, \sqrt{-16 \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 16 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 16} - 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) - 2 \, \arctan \left (-\sqrt{3} + 2 \, \sqrt{\sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1} - 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + 2 \, \arctan \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + \frac{1}{2} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) - \frac{1}{2} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) + \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) - \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} - 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 [6]{\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 = 1.18374, size = 352, normalized size = 0.88 \begin{align*} -\sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right )}\right ) - \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} - 1\right )}\right ) - \frac{1}{2} \, \sqrt{3} \log \left (\sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) + \frac{1}{2} \, \sqrt{3} \log \left (-\sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) + \arctan \left (\sqrt{3} + 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + \arctan \left (-\sqrt{3} + 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + 2 \, \arctan \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + \frac{1}{2} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) - \frac{1}{2} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) + \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) - \log \left ({\left | \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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