Optimal. Leaf size=224 \[ -\frac{(1-x)^{5/6} (x+1)^{7/6}}{2 x^2}-\frac{(1-x)^{5/6} \sqrt [6]{x+1}}{6 x}+\frac{1}{36} \log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{1-x}}-\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-\frac{1}{36} \log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}}{\sqrt{3}}\right )}{6 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1}{\sqrt{3}}\right )}{6 \sqrt{3}}-\frac{1}{9} \tanh ^{-1}\left (\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}\right ) \]
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Rubi [A] time = 0.17509, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6126, 96, 94, 93, 210, 634, 618, 204, 628, 206} \[ -\frac{(1-x)^{5/6} (x+1)^{7/6}}{2 x^2}-\frac{(1-x)^{5/6} \sqrt [6]{x+1}}{6 x}+\frac{1}{36} \log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{1-x}}-\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-\frac{1}{36} \log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}}{\sqrt{3}}\right )}{6 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1}{\sqrt{3}}\right )}{6 \sqrt{3}}-\frac{1}{9} \tanh ^{-1}\left (\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}\right ) \]
Antiderivative was successfully verified.
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Rule 6126
Rule 96
Rule 94
Rule 93
Rule 210
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{3} \tanh ^{-1}(x)}}{x^3} \, dx &=\int \frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x} x^3} \, dx\\ &=-\frac{(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}+\frac{1}{6} \int \frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x} x^2} \, dx\\ &=-\frac{(1-x)^{5/6} \sqrt [6]{1+x}}{6 x}-\frac{(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}+\frac{1}{18} \int \frac{1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx\\ &=-\frac{(1-x)^{5/6} \sqrt [6]{1+x}}{6 x}-\frac{(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+x^6} \, dx,x,\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\\ &=-\frac{(1-x)^{5/6} \sqrt [6]{1+x}}{6 x}-\frac{(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}-\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac{1}{9} \operatorname{Subst}\left (\int \frac{1-\frac{x}{2}}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac{1}{9} \operatorname{Subst}\left (\int \frac{1+\frac{x}{2}}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\\ &=-\frac{(1-x)^{5/6} \sqrt [6]{1+x}}{6 x}-\frac{(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}-\frac{1}{9} \tanh ^{-1}\left (\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac{1}{36} \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac{1}{36} \operatorname{Subst}\left (\int \frac{1+2 x}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac{1}{12} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac{1}{12} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\\ &=-\frac{(1-x)^{5/6} \sqrt [6]{1+x}}{6 x}-\frac{(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}-\frac{1}{9} \tanh ^{-1}\left (\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac{1}{36} \log \left (1-\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac{1}{36} \log \left (1+\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+\frac{2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\\ &=-\frac{(1-x)^{5/6} \sqrt [6]{1+x}}{6 x}-\frac{(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt{3}}\right )}{6 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1+\frac{2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt{3}}\right )}{6 \sqrt{3}}-\frac{1}{9} \tanh ^{-1}\left (\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac{1}{36} \log \left (1-\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac{1}{36} \log \left (1+\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0145863, size = 60, normalized size = 0.27 \[ -\frac{(1-x)^{5/6} \left (2 x^2 \text{Hypergeometric2F1}\left (\frac{5}{6},1,\frac{11}{6},\frac{1-x}{x+1}\right )+5 \left (4 x^2+7 x+3\right )\right )}{30 x^2 (x+1)^{5/6}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\sqrt [3]{{(1+x){\frac{1}{\sqrt{-{x}^{2}+1}}}}}}\, 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{\left (\frac{x + 1}{\sqrt{-x^{2} + 1}}\right )^{\frac{1}{3}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73425, size = 657, normalized size = 2.93 \begin{align*} -\frac{2 \, \sqrt{3} x^{2} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + 2 \, \sqrt{3} x^{2} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + x^{2} \log \left (\left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}} + \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} + 1\right ) - x^{2} \log \left (\left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}} - \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} + 1\right ) + 2 \, x^{2} \log \left (\left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} + 1\right ) - 2 \, x^{2} \log \left (\left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} - 1\right ) - 6 \,{\left (4 \, x^{2} - x - 3\right )} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}}}{36 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{x + 1}{\sqrt{-x^{2} + 1}}\right )^{\frac{1}{3}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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