Optimal. Leaf size=111 \[ -\frac{(1-i x)^{2/3} \sqrt [3]{1+i x}}{x}+i \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )-\frac{1}{3} i \log (x)+\frac{2 i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{1-i x}}{\sqrt{3} \sqrt [3]{1+i x}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0292463, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5062, 94, 91} \[ -\frac{(1-i x)^{2/3} \sqrt [3]{1+i x}}{x}+i \log \left (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )-\frac{1}{3} i \log (x)+\frac{2 i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{1-i x}}{\sqrt{3} \sqrt [3]{1+i x}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 94
Rule 91
Rubi steps
\begin{align*} \int \frac{e^{\frac{2}{3} i \tan ^{-1}(x)}}{x^2} \, dx &=\int \frac{\sqrt [3]{1+i x}}{\sqrt [3]{1-i x} x^2} \, dx\\ &=-\frac{(1-i x)^{2/3} \sqrt [3]{1+i x}}{x}+\frac{2}{3} i \int \frac{1}{\sqrt [3]{1-i x} (1+i x)^{2/3} x} \, dx\\ &=-\frac{(1-i x)^{2/3} \sqrt [3]{1+i x}}{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 (\sqrt [3]{1-i x}-\sqrt [3]{1+i x}\right )-\frac{1}{3} i \log (x)\\ \end{align*}
Mathematica [C] time = 0.0115087, size = 59, normalized size = 0.53 \[ -\frac{i (1-i x)^{2/3} \left (x \text{Hypergeometric2F1}\left (\frac{2}{3},1,\frac{5}{3},\frac{x+i}{-x+i}\right )+x-i\right )}{(1+i x)^{2/3} x} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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 \frac{\left (\frac{i \, x + 1}{\sqrt{x^{2} + 1}}\right )^{\frac{2}{3}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69989, size = 352, normalized size = 3.17 \begin{align*} \frac{{\left (\sqrt{3} x - i \, x\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 ) -{\left (\sqrt{3} x + i \, x\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 ) + 2 i \, x \log \left (\left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{2}{3}} - 1\right ) - 3 \,{\left (-i \, x + 1\right )} \left (\frac{i \, \sqrt{x^{2} + 1}}{x + i}\right )^{\frac{2}{3}}}{3 \, x} \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 \frac{\left (\frac{i \, x + 1}{\sqrt{x^{2} + 1}}\right )^{\frac{2}{3}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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