Optimal. Leaf size=163 \[ \frac {3}{2} \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )+\frac {3}{2} \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )+\frac {1}{2} \log (1+i x)-\frac {\log (x)}{2}+\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5062, 105, 60, 91} \[ \frac {3}{2} \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )+\frac {3}{2} \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )+\frac {1}{2} \log (1+i x)-\frac {\log (x)}{2}+\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right ) \]
Antiderivative was successfully verified.
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Rule 60
Rule 91
Rule 105
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{\frac {2}{3} i \tan ^{-1}(x)}}{x} \, dx &=\int \frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x} x} \, dx\\ &=i \int \frac {1}{\sqrt [3]{1-i x} (1+i x)^{2/3}} \, dx+\int \frac {1}{\sqrt [3]{1-i x} (1+i x)^{2/3} x} \, dx\\ &=\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-i x}}{\sqrt {3} \sqrt [3]{1+i x}}\right )+\frac {3}{2} \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right )+\frac {3}{2} \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )+\frac {1}{2} \log (1+i x)-\frac {\log (x)}{2}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 90, normalized size = 0.55 \[ -\frac {3 (1-i x)^{2/3} \left (\sqrt [3]{2} (1+i x)^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {1}{2}-\frac {i x}{2}\right )+2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {x+i}{i-x}\right )\right )}{4 (1+i x)^{2/3}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 147, normalized size = 0.90 \[ \frac {1}{2} \, {\left (i \, \sqrt {3} - 1\right )} \log \left (\frac {\sqrt {3} {\left (i \, x - 1\right )} + x + 2 i \, \sqrt {x^{2} + 1} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + i}{2 \, x + 2 i}\right ) + \frac {1}{2} \, {\left (-i \, \sqrt {3} - 1\right )} \log \left (\frac {\sqrt {3} {\left (-i \, x + 1\right )} + x + 2 i \, \sqrt {x^{2} + 1} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + i}{2 \, x + 2 i}\right ) + \log \left (-\frac {x - i \, \sqrt {x^{2} + 1} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + i}{x + i}\right ) \]
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 \[ \int \frac {\left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {2}{3}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )^{\frac {2}{3}}}{x}\, dx \]
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 \[ \int \frac {\left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {2}{3}}}{x}\,{d x} \]
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 \[ \int \frac {{\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{2/3}}{x} \,d x \]
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 \[ \int \frac {\left (\frac {i \left (x - i\right )}{\sqrt {x^{2} + 1}}\right )^{\frac {2}{3}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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