Optimal. Leaf size=140 \[ \frac {1}{2} (1-i x)^{2/3} (1+i x)^{4/3}+\frac {1}{3} (1-i x)^{2/3} \sqrt [3]{1+i x}-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )-\frac {1}{9} \log (1+i x)-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5062, 80, 50, 60} \[ \frac {1}{2} (1-i x)^{2/3} (1+i x)^{4/3}+\frac {1}{3} (1-i x)^{2/3} \sqrt [3]{1+i x}-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )-\frac {1}{9} \log (1+i x)-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 60
Rule 80
Rule 5062
Rubi steps
\begin {align*} \int e^{\frac {2}{3} i \tan ^{-1}(x)} x \, dx &=\int \frac {\sqrt [3]{1+i x} x}{\sqrt [3]{1-i x}} \, dx\\ &=\frac {1}{2} (1-i x)^{2/3} (1+i x)^{4/3}-\frac {1}{3} i \int \frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}} \, dx\\ &=\frac {1}{3} (1-i x)^{2/3} \sqrt [3]{1+i x}+\frac {1}{2} (1-i x)^{2/3} (1+i x)^{4/3}-\frac {2}{9} i \int \frac {1}{\sqrt [3]{1-i x} (1+i x)^{2/3}} \, dx\\ &=\frac {1}{3} (1-i x)^{2/3} \sqrt [3]{1+i x}+\frac {1}{2} (1-i x)^{2/3} (1+i x)^{4/3}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )}{3 \sqrt {3}}-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )-\frac {1}{9} \log (1+i x)\\ \end {align*}
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Mathematica [C] time = 0.02, size = 54, normalized size = 0.39 \[ \frac {1}{2} (1-i x)^{2/3} \left (\sqrt [3]{2} \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {1}{2}-\frac {i x}{2}\right )+(1+i x)^{4/3}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.62, size = 116, normalized size = 0.83 \[ -\frac {1}{9} \, {\left (i \, \sqrt {3} - 1\right )} \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {2}{3}} + \frac {1}{2} i \, \sqrt {3} - \frac {1}{2}\right ) - \frac {1}{9} \, {\left (-i \, \sqrt {3} - 1\right )} \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {2}{3}} - \frac {1}{2} i \, \sqrt {3} - \frac {1}{2}\right ) + \frac {1}{6} \, {\left (3 \, x^{2} - 2 i \, x + 5\right )} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {2}{3}} - \frac {2}{9} \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {2}{3}} + 1\right ) \]
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 \[ \int x \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )^{\frac {2}{3}} x\, dx \]
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 \[ \int x \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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