Optimal. Leaf size=116 \[ i (1-i x)^{2/3} \sqrt [3]{1+i x}-i \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )-\frac {1}{3} i \log (1+i x)-\frac {2 i \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5061, 50, 60} \[ i (1-i x)^{2/3} \sqrt [3]{1+i x}-i \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )-\frac {1}{3} i \log (1+i x)-\frac {2 i \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 60
Rule 5061
Rubi steps
\begin {align*} \int e^{\frac {2}{3} i \tan ^{-1}(x)} \, dx &=\int \frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}} \, dx\\ &=i (1-i x)^{2/3} \sqrt [3]{1+i x}+\frac {2}{3} \int \frac {1}{\sqrt [3]{1-i x} (1+i x)^{2/3}} \, dx\\ &=i (1-i x)^{2/3} \sqrt [3]{1+i x}-\frac {2 i \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )}{\sqrt {3}}-i \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )-\frac {1}{3} i \log (1+i x)\\ \end {align*}
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Mathematica [C] time = 0.02, size = 34, normalized size = 0.29 \[ -\frac {3}{2} i e^{\frac {8}{3} i \tan ^{-1}(x)} \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};-e^{2 i \tan ^{-1}(x)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.63, size = 107, normalized size = 0.92 \[ \frac {1}{3} \, {\left (\sqrt {3} + i\right )} \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {2}{3}} + \frac {1}{2} i \, \sqrt {3} - \frac {1}{2}\right ) - \frac {1}{3} \, {\left (\sqrt {3} - i\right )} \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {2}{3}} - \frac {1}{2} i \, \sqrt {3} - \frac {1}{2}\right ) + \frac {1}{3} \, {\left (3 \, x + 3 i\right )} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {2}{3}} - \frac {2}{3} i \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {2}{3}} + 1\right ) \]
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 \[ \int \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )^{\frac {2}{3}}\, dx \]
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 \[ \int \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {2}{3}}\,{d x} \]
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 \[ \int {\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{2/3} \,d x \]
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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