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 = 0.0205023, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 5061
Rule 50
Rule 60
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*}
Mathematica [C] time = 0.0172518, size = 34, normalized size = 0.29 \[ -\frac{3}{2} i e^{\frac{8}{3} i \tan ^{-1}(x)} \text{Hypergeometric2F1}\left (\frac{4}{3},2,\frac{7}{3},-e^{2 i \tan ^{-1}(x)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \left ({(1+ix){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) ^{{\frac{2}{3}}}\, 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 \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 = 1.70289, size = 346, normalized size = 2.98 \begin{align*} \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 ) \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 \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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