Optimal. Leaf size=278 \[ \frac{1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac{1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac{\log \left (\frac{\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac{\sqrt{3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{12 \sqrt{3}}-\frac{\log \left (\frac{\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac{\sqrt{3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{12 \sqrt{3}}-\frac{1}{18} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac{1}{18} \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right ) \]
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Rubi [A] time = 0.338368, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {5062, 80, 50, 63, 331, 295, 634, 618, 204, 628, 203} \[ \frac{1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac{1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac{\log \left (\frac{\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac{\sqrt{3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{12 \sqrt{3}}-\frac{\log \left (\frac{\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac{\sqrt{3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{12 \sqrt{3}}-\frac{1}{18} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac{1}{18} \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right ) \]
Antiderivative was successfully verified.
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Rule 5062
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} i \tan ^{-1}(x)} x \, dx &=\int \frac{\sqrt [6]{1+i x} x}{\sqrt [6]{1-i x}} \, dx\\ &=\frac{1}{2} (1-i x)^{5/6} (1+i x)^{7/6}-\frac{1}{6} i \int \frac{\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}} \, dx\\ &=\frac{1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac{1}{2} (1-i x)^{5/6} (1+i x)^{7/6}-\frac{1}{18} i \int \frac{1}{\sqrt [6]{1-i x} (1+i x)^{5/6}} \, dx\\ &=\frac{1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac{1}{2} (1-i x)^{5/6} (1+i 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-i x}\right )\\ &=\frac{1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac{1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^4}{1+x^6} \, dx,x,\frac{\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\\ &=\frac{1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac{1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [6]{1-i x}}{\sqrt [6]{1+i 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-i x}}{\sqrt [6]{1+i 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-i x}}{\sqrt [6]{1+i x}}\right )\\ &=\frac{1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac{1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac{1}{36} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac{1}{36} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac{\operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-i x}}{\sqrt [6]{1+i 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-i x}}{\sqrt [6]{1+i x}}\right )}{12 \sqrt{3}}\\ &=\frac{1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac{1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac{\log \left (1+\frac{\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac{\sqrt{3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{12 \sqrt{3}}-\frac{\log \left (1+\frac{\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac{\sqrt{3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i 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-i x}}{\sqrt [6]{1+i x}}\right )-\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+\frac{2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\\ &=\frac{1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac{1}{2} (1-i x)^{5/6} (1+i x)^{7/6}-\frac{1}{18} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac{1}{18} \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac{\log \left (1+\frac{\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac{\sqrt{3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{12 \sqrt{3}}-\frac{\log \left (1+\frac{\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac{\sqrt{3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{12 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.022786, size = 57, normalized size = 0.21 \[ \frac{1}{10} (1-i x)^{5/6} \left (2 \sqrt [6]{2} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{5}{6},\frac{11}{6},\frac{1}{2}-\frac{i x}{2}\right )+5 (1+i x)^{7/6}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{{(1+ix){\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{i \, 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 = 1.78683, size = 645, normalized size = 2.32 \begin{align*} -\frac{1}{36} \,{\left (\sqrt{3} + i\right )} \log \left (\frac{1}{2} \, \sqrt{3} + \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} + \frac{1}{2} i\right ) - \frac{1}{36} \,{\left (\sqrt{3} - i\right )} \log \left (\frac{1}{2} \, \sqrt{3} + \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} - \frac{1}{2} i\right ) + \frac{1}{36} \,{\left (\sqrt{3} - i\right )} \log \left (-\frac{1}{2} \, \sqrt{3} + \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} + \frac{1}{2} i\right ) + \frac{1}{36} \,{\left (\sqrt{3} + i\right )} \log \left (-\frac{1}{2} \, \sqrt{3} + \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} - \frac{1}{2} i\right ) + \frac{1}{6} \,{\left (3 \, x^{2} - i \, x + 4\right )} \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} - \frac{1}{18} i \, \log \left (\left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} + i\right ) + \frac{1}{18} i \, \log \left (\left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{1}{3}} - i\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{i \, 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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