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 = 0.34, antiderivative size = 278, normalized size of antiderivative = 1.00, 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 50
Rule 63
Rule 80
Rule 203
Rule 204
Rule 295
Rule 331
Rule 618
Rule 628
Rule 634
Rule 5062
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*}
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Mathematica [C] time = 0.02, size = 57, normalized size = 0.21 \[ \frac {1}{10} (1-i x)^{5/6} \left (2 \sqrt [6]{2} \, _2F_1\left (-\frac {1}{6},\frac {5}{6};\frac {11}{6};\frac {1}{2}-\frac {i x}{2}\right )+5 (1+i x)^{7/6}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 195, normalized size = 0.70 \[ -\frac {1}{36} \, {\left (\sqrt {3} + i\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) - \frac {1}{36} \, {\left (\sqrt {3} - i\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + \frac {1}{36} \, {\left (\sqrt {3} - i\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) + \frac {1}{36} \, {\left (\sqrt {3} + i\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + \frac {1}{6} \, {\left (3 \, x^{2} - i \, x + 4\right )} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{18} i \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + i\right ) + \frac {1}{18} i \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )^{\frac {1}{3}} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{1/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt [3]{\frac {i \left (x - i\right )}{\sqrt {x^{2} + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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