Optimal. Leaf size=202 \[ -(1-x)^{5/6} \sqrt [6]{x+1}-\frac{\log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{2 \sqrt{3}}+\frac{\log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{2 \sqrt{3}}-\frac{2}{3} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac{1}{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt{3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32911, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.25, Rules used = {6125, 50, 63, 331, 295, 634, 618, 204, 628, 203} \[ -(1-x)^{5/6} \sqrt [6]{x+1}-\frac{\log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{2 \sqrt{3}}+\frac{\log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{2 \sqrt{3}}-\frac{2}{3} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac{1}{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6125
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)} \, dx &=\int \frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}+\frac{1}{3} \int \frac{1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}-2 \operatorname{Subst}\left (\int \frac{x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-x}\right )\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}-2 \operatorname{Subst}\left (\int \frac{x^4}{1+x^6} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{2}{3} \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{2}{3} \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 )\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}-\frac{2}{3} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{1}{6} \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}{6} \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 )}{2 \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 )}{2 \sqrt{3}}\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}-\frac{2}{3} \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 )}{2 \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 )}{2 \sqrt{3}}+\frac{1}{3} \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}{3} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}-\frac{2}{3} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{1}{3} \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 )}{2 \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 )}{2 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0433318, size = 39, normalized size = 0.19 \[ 2 e^{\frac{1}{3} \tanh ^{-1}(x)} \left (\text{Hypergeometric2F1}\left (\frac{1}{6},1,\frac{7}{6},-e^{2 \tanh ^{-1}(x)}\right )-\frac{1}{e^{2 \tanh ^{-1}(x)}+1}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.035, size = 0, normalized size = 0. \begin{align*} \int \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 \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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.79535, size = 791, normalized size = 3.92 \begin{align*} \frac{1}{6} \, \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} \, \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 ) +{\left (x - 1\right )} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}} - \frac{2}{3} \, \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{2}{3} \, \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{2}{3} \, \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{\frac{x + 1}{\sqrt{1 - x^{2}}}}\, 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 \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.
[In]
[Out]