Optimal. Leaf size=109 \[ -\frac{i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{12 \sqrt [3]{2}}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt{3}}+\frac{i \log (2+3 i x)}{12 \sqrt [3]{2}} \]
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Rubi [A] time = 0.0162252, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {751} \[ -\frac{i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{12 \sqrt [3]{2}}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt{3}}+\frac{i \log (2+3 i x)}{12 \sqrt [3]{2}} \]
Antiderivative was successfully verified.
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Rule 751
Rubi steps
\begin{align*} \int \frac{1}{(2+3 i x) \sqrt [3]{4-27 x^2}} \, dx &=\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt{3}}+\frac{i \log (2+3 i x)}{12 \sqrt [3]{2}}-\frac{i \log \left (-54+81 i x+27\ 2^{2/3} \sqrt [3]{4-27 x^2}\right )}{12 \sqrt [3]{2}}\\ \end{align*}
Mathematica [C] time = 0.0829706, size = 125, normalized size = 1.15 \[ \frac{i \sqrt [3]{\frac{2 \sqrt{3}-9 x}{-3 x+2 i}} \sqrt [3]{\frac{9 x+2 \sqrt{3}}{3 x-2 i}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2 \left (3 i+\sqrt{3}\right )}{6 i-9 x},\frac{2 \left (-3 i+\sqrt{3}\right )}{9 x-6 i}\right )}{2\ 3^{2/3} \sqrt [3]{4-27 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.411, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{2+3\,ix}{\frac{1}{\sqrt [3]{-27\,{x}^{2}+4}}}}\, 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{1}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 i \, x + 2\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{4 - 27 x^{2}} \left (3 i x + 2\right )}\, dx \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{1}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 i \, x + 2\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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