Optimal. Leaf size=224 \[ -\frac{1}{2} (1-x)^{5/6} (x+1)^{7/6}-\frac{1}{6} (1-x)^{5/6} \sqrt [6]{x+1}-\frac{\log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{12 \sqrt{3}}+\frac{\log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{12 \sqrt{3}}-\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac{1}{18} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac{1}{18} \tan ^{-1}\left (\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt{3}\right ) \]
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Rubi [A] time = 0.34011, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.1, Rules used = {6126, 80, 50, 63, 331, 295, 634, 618, 204, 628, 203} \[ -\frac{1}{2} (1-x)^{5/6} (x+1)^{7/6}-\frac{1}{6} (1-x)^{5/6} \sqrt [6]{x+1}-\frac{\log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{12 \sqrt{3}}+\frac{\log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{12 \sqrt{3}}-\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac{1}{18} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac{1}{18} \tan ^{-1}\left (\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
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Rule 6126
Rule 80
Rule 50
Rule 63
Rule 331
Rule 295
Rule 634
Rule 618
Rule 204
Rule 628
Rule 203
Rubi steps
\begin{align*} \int e^{\frac{1}{3} \tanh ^{-1}(x)} x \, dx &=\int \frac{x \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx\\ &=-\frac{1}{2} (1-x)^{5/6} (1+x)^{7/6}+\frac{1}{6} \int \frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx\\ &=-\frac{1}{6} (1-x)^{5/6} \sqrt [6]{1+x}-\frac{1}{2} (1-x)^{5/6} (1+x)^{7/6}+\frac{1}{18} \int \frac{1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx\\ &=-\frac{1}{6} (1-x)^{5/6} \sqrt [6]{1+x}-\frac{1}{2} (1-x)^{5/6} (1+x)^{7/6}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-x}\right )\\ &=-\frac{1}{6} (1-x)^{5/6} \sqrt [6]{1+x}-\frac{1}{2} (1-x)^{5/6} (1+x)^{7/6}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^4}{1+x^6} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-\frac{1}{6} (1-x)^{5/6} \sqrt [6]{1+x}-\frac{1}{2} (1-x)^{5/6} (1+x)^{7/6}-\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{-\frac{1}{2}+\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{1}{9} \operatorname{Subst}\left (\int \frac{-\frac{1}{2}-\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-\frac{1}{6} (1-x)^{5/6} \sqrt [6]{1+x}-\frac{1}{2} (1-x)^{5/6} (1+x)^{7/6}-\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{1}{36} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{1}{36} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{\operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt{3}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt{3}}\\ &=-\frac{1}{6} (1-x)^{5/6} \sqrt [6]{1+x}-\frac{1}{2} (1-x)^{5/6} (1+x)^{7/6}-\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{\log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt{3}}+\frac{\log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt{3}}+\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-\frac{1}{6} (1-x)^{5/6} \sqrt [6]{1+x}-\frac{1}{2} (1-x)^{5/6} (1+x)^{7/6}-\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{1}{18} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{1}{18} \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{\log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt{3}}+\frac{\log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0132472, size = 49, normalized size = 0.22 \[ -\frac{1}{10} (1-x)^{5/6} \left (2 \sqrt [6]{2} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{5}{6},\frac{11}{6},\frac{1-x}{2}\right )+5 (x+1)^{7/6}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{{(1+x){\frac{1}{\sqrt{-{x}^{2}+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 x \left (\frac{x + 1}{\sqrt{-x^{2} + 1}}\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73803, size = 810, normalized size = 3.62 \begin{align*} \frac{1}{36} \, \sqrt{3} \log \left (4 \, \sqrt{3} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} + 4 \, \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}} + 4\right ) - \frac{1}{36} \, \sqrt{3} \log \left (-4 \, \sqrt{3} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} + 4 \, \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}} + 4\right ) + \frac{1}{6} \,{\left (3 \, x^{2} + x - 4\right )} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} - \frac{1}{9} \, \arctan \left (\sqrt{3} + \sqrt{-4 \, \sqrt{3} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} + 4 \, \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}} + 4} - 2 \, \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}}\right ) - \frac{1}{9} \, \arctan \left (-\sqrt{3} + 2 \, \sqrt{\sqrt{3} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} + \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}} + 1} - 2 \, \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}}\right ) + \frac{1}{9} \, \arctan \left (\left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}}\right ) \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 x \left (\frac{x + 1}{\sqrt{-x^{2} + 1}}\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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