Optimal. Leaf size=262 \[ i (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {i \log \left (\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{2 \sqrt {3}}-\frac {i \log \left (\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{2 \sqrt {3}}-\frac {1}{3} i \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{3} i \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {2}{3} i \tan ^{-1}\left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right ) \]
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Rubi [A] time = 0.31, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5061, 50, 63, 331, 295, 634, 618, 204, 628, 203} \[ i (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {i \log \left (\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{2 \sqrt {3}}-\frac {i \log \left (\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{2 \sqrt {3}}-\frac {1}{3} i \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{3} i \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {2}{3} i \tan ^{-1}\left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 204
Rule 295
Rule 331
Rule 618
Rule 628
Rule 634
Rule 5061
Rubi steps
\begin {align*} \int e^{\frac {1}{3} i \tan ^{-1}(x)} \, dx &=\int \frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}} \, dx\\ &=i (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {1}{3} \int \frac {1}{\sqrt [6]{1-i x} (1+i x)^{5/6}} \, dx\\ &=i (1-i x)^{5/6} \sqrt [6]{1+i x}+2 i \operatorname {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-i x}\right )\\ &=i (1-i x)^{5/6} \sqrt [6]{1+i x}+2 i \operatorname {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\\ &=i (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {2}{3} i \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {2}{3} i \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 {2}{3} i \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 )\\ &=i (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {2}{3} i \tan ^{-1}\left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{6} i \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}{6} i \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 {i \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 )}{2 \sqrt {3}}-\frac {i \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 )}{2 \sqrt {3}}\\ &=i (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {2}{3} i \tan ^{-1}\left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {i \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 )}{2 \sqrt {3}}-\frac {i \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 )}{2 \sqrt {3}}-\frac {1}{3} i \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}{3} i \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 )\\ &=i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {1}{3} i \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{3} i \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {2}{3} i \tan ^{-1}\left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {i \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 )}{2 \sqrt {3}}-\frac {i \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 )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 34, normalized size = 0.13 \[ -\frac {12}{7} i e^{\frac {7}{3} i \tan ^{-1}(x)} \, _2F_1\left (\frac {7}{6},2;\frac {13}{6};-e^{2 i \tan ^{-1}(x)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 198, normalized size = 0.76 \[ \frac {1}{6} \, {\left (-i \, \sqrt {3} + 1\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) + \frac {1}{6} \, {\left (-i \, \sqrt {3} - 1\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + \frac {1}{6} \, {\left (i \, \sqrt {3} + 1\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) + \frac {1}{6} \, {\left (i \, \sqrt {3} - 1\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + \frac {1}{6} \, {\left (6 \, x + 6 i\right )} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{3} \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + i\right ) - \frac {1}{3} \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - i\right ) \]
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 \[ \int \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )^{\frac {1}{3}}\, dx \]
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 \[ \int \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}}\,{d x} \]
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 \[ \int {\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{1/3} \,d x \]
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 \[ \int \sqrt [3]{\frac {i x + 1}{\sqrt {x^{2} + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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