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 = 0.0359681, antiderivative size = 140, normalized size of antiderivative = 1., 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 5062
Rule 80
Rule 50
Rule 60
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*}
Mathematica [C] time = 0.0198766, size = 54, normalized size = 0.39 \[ \frac{1}{2} (1-i x)^{2/3} \left (\sqrt [3]{2} \text{Hypergeometric2F1}\left (-\frac{1}{3},\frac{2}{3},\frac{5}{3},\frac{1}{2}-\frac{i x}{2}\right )+(1+i x)^{4/3}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int \left ({(1+ix){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) ^{{\frac{2}{3}}}x\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (\frac{i \, x + 1}{\sqrt{x^{2} + 1}}\right )^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6965, size = 362, normalized size = 2.59 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (\frac{i \, x + 1}{\sqrt{x^{2} + 1}}\right )^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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