Optimal. Leaf size=346 \[ -\frac{1}{2} \sqrt{3} \log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )+\frac{1}{2} \sqrt{3} \log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )+\frac{1}{2} \log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{1-x}}-\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-\frac{1}{2} \log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-2 \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\tan ^{-1}\left (\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt{3}\right )+\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}}{\sqrt{3}}\right )-\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1}{\sqrt{3}}\right )-2 \tanh ^{-1}\left (\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.454681, antiderivative size = 346, 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 = {6126, 105, 63, 331, 295, 634, 618, 204, 628, 203, 93, 210, 206} \[ -\frac{1}{2} \sqrt{3} \log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )+\frac{1}{2} \sqrt{3} \log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )+\frac{1}{2} \log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{1-x}}-\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-\frac{1}{2} \log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-2 \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\tan ^{-1}\left (\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt{3}\right )+\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}}{\sqrt{3}}\right )-\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1}{\sqrt{3}}\right )-2 \tanh ^{-1}\left (\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6126
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} \tanh ^{-1}(x)}}{x} \, dx &=\int \frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x} x} \, dx\\ &=\int \frac{1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx+\int \frac{1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx\\ &=-\left (6 \operatorname{Subst}\left (\int \frac{x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-x}\right )\right )+6 \operatorname{Subst}\left (\int \frac{1}{-1+x^6} \, dx,x,\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\right )-2 \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 )-2 \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 )-6 \operatorname{Subst}\left (\int \frac{x^4}{1+x^6} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-2 \tanh ^{-1}\left (\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac{1}{2} \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}{2} \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{3}{2} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{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]{1-x}}{\sqrt [6]{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]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-2 \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-2 \tanh ^{-1}\left (\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac{1}{2} \log \left (1-\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac{1}{2} \log \left (1+\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac{1}{2} \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}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+\frac{2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+3 \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}{2} \sqrt{3} \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 )+\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]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-2 \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt{3}}\right )-\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt{3}}\right )-2 \tanh ^{-1}\left (\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac{1}{2} \sqrt{3} \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{1}{2} \sqrt{3} \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{1}{2} \log \left (1-\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac{1}{2} \log \left (1+\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )+\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-2 \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt{3}}\right )-\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt{3}}\right )-2 \tanh ^{-1}\left (\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac{1}{2} \sqrt{3} \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{1}{2} \sqrt{3} \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{1}{2} \log \left (1-\frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac{1}{2} \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 = 0.0255055, size = 74, normalized size = 0.21 \[ -\frac{3 (1-x)^{5/6} \left (\sqrt [6]{2} (x+1)^{5/6} \text{Hypergeometric2F1}\left (\frac{5}{6},\frac{5}{6},\frac{11}{6},\frac{1-x}{2}\right )+2 \text{Hypergeometric2F1}\left (\frac{5}{6},1,\frac{11}{6},\frac{1-x}{x+1}\right )\right )}{5 (x+1)^{5/6}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt [3]{{(1+x){\frac{1}{\sqrt{-{x}^{2}+1}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.9226, size = 1268, normalized size = 3.66 \begin{align*} -\sqrt{3} \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 ) - \sqrt{3} \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 ) + \frac{1}{2} \, \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}{2} \, \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 ) - 2 \, \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 ) - 2 \, \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 ) + 2 \, \arctan \left (\left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}}\right ) - \frac{1}{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 ) + \frac{1}{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 ) - \log \left (\left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} + 1\right ) + \log \left (\left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{\frac{x + 1}{\sqrt{1 - x^{2}}}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]