Optimal. Leaf size=109 \[ \frac {i \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 i x)}{\sqrt {3} \sqrt [3]{4-27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt {3}}+\frac {i \log (2+3 i x)}{12 \sqrt [3]{2}}-\frac {i \log \left (-54+81 i x+27\ 2^{2/3} \sqrt [3]{4-27 x^2}\right )}{12 \sqrt [3]{2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 = {765}
\begin {gather*} \frac {i \text {ArcTan}\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 \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{12 \sqrt [3]{2}}+\frac {i \log (2+3 i x)}{12 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
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*}
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Mathematica [A]
time = 0.30, size = 166, normalized size = 1.52 \begin {gather*} -\frac {i \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{4-27 x^2}}{2 \sqrt [3]{2}-3 i \sqrt [3]{2} x+\sqrt [3]{4-27 x^2}}\right )+2 \log \left (-2 \sqrt [3]{2}+3 i \sqrt [3]{2} x+2 \sqrt [3]{4-27 x^2}\right )-\log \left (-4 2^{2/3}+12 i 2^{2/3} x+9\ 2^{2/3} x^2+2 (-2+3 i x) \sqrt [3]{8-54 x^2}-4 \left (4-27 x^2\right )^{2/3}\right )\right )}{36 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (3 i x +2\right ) \left (-27 x^{2}+4\right )^{\frac {1}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} - i \int \frac {1}{3 x \sqrt [3]{4 - 27 x^{2}} - 2 i \sqrt [3]{4 - 27 x^{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (2+x\,3{}\mathrm {i}\right )\,{\left (4-27\,x^2\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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