Optimal. Leaf size=375 \[ \frac {i \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+\frac {i \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}-\frac {i \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+i \sqrt {3}}} \]
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Rubi [A]
time = 0.16, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1389, 298, 31,
648, 631, 210, 642} \begin {gather*} \frac {i \text {ArcTan}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \text {ArcTan}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}-\frac {i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+i \sqrt {3}}}+\frac {i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 298
Rule 631
Rule 642
Rule 648
Rule 1389
Rubi steps
\begin {align*} \int \frac {x}{1-x^3+x^6} \, dx &=-\frac {i \int \frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}+x^3} \, dx}{\sqrt {3}}+\frac {i \int \frac {x}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}+x^3} \, dx}{\sqrt {3}}\\ &=\frac {i \int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+x} \, dx}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \int \frac {-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+x}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+x^2} \, dx}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}+x} \, dx}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+\frac {i \int \frac {-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}+x}{\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+x^2} \, dx}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}\\ &=\frac {i \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+\frac {i \int \frac {1}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+x^2} \, dx}{2 \sqrt {3}}-\frac {i \int \frac {1}{\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+x^2} \, dx}{2 \sqrt {3}}-\frac {i \int \frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+2 x}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+x^2} \, dx}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \int \frac {\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}+2 x}{\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+x^2} \, dx}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+i \sqrt {3}}}\\ &=\frac {i \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}-\frac {i \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+i \sqrt {3}}}-\frac {i \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}+\frac {i \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}\\ &=\frac {i \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+\frac {i \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}-\frac {i \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+i \sqrt {3}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.01, size = 40, normalized size = 0.11 \begin {gather*} \frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.02, size = 38, normalized size = 0.10
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}\) | \(38\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}\) | \(38\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 994 vs.
\(2 (241) = 482\).
time = 0.45, size = 994, normalized size = 2.65 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 26, normalized size = 0.07 \begin {gather*} \operatorname {RootSum} {\left (19683 t^{6} - 243 t^{3} + 1, \left ( t \mapsto t \log {\left (6561 t^{5} - 27 t^{2} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 815 vs. \(2 (241) = 482\).
time = 4.18, size = 815, normalized size = 2.17 \begin {gather*} -\frac {1}{9} \, {\left (\sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{5} - 10 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{3} \sin \left (\frac {4}{9} \, \pi \right )^{2} + 5 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right )^{4} - 5 \, \cos \left (\frac {4}{9} \, \pi \right )^{4} \sin \left (\frac {4}{9} \, \pi \right ) + 10 \, \cos \left (\frac {4}{9} \, \pi \right )^{2} \sin \left (\frac {4}{9} \, \pi \right )^{3} - \sin \left (\frac {4}{9} \, \pi \right )^{5} - \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{2} + \sqrt {3} \sin \left (\frac {4}{9} \, \pi \right )^{2} + 2 \, \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right )\right )} \arctan \left (\frac {{\left (-i \, \sqrt {3} - 1\right )} \cos \left (\frac {4}{9} \, \pi \right ) + 2 \, x}{-{\left (-i \, \sqrt {3} - 1\right )} \sin \left (\frac {4}{9} \, \pi \right )}\right ) - \frac {1}{9} \, {\left (\sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{5} - 10 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{3} \sin \left (\frac {2}{9} \, \pi \right )^{2} + 5 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right )^{4} - 5 \, \cos \left (\frac {2}{9} \, \pi \right )^{4} \sin \left (\frac {2}{9} \, \pi \right ) + 10 \, \cos \left (\frac {2}{9} \, \pi \right )^{2} \sin \left (\frac {2}{9} \, \pi \right )^{3} - \sin \left (\frac {2}{9} \, \pi \right )^{5} - \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{2} + \sqrt {3} \sin \left (\frac {2}{9} \, \pi \right )^{2} + 2 \, \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right )\right )} \arctan \left (\frac {{\left (-i \, \sqrt {3} - 1\right )} \cos \left (\frac {2}{9} \, \pi \right ) + 2 \, x}{-{\left (-i \, \sqrt {3} - 1\right )} \sin \left (\frac {2}{9} \, \pi \right )}\right ) + \frac {1}{9} \, {\left (\sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{5} - 10 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{3} \sin \left (\frac {1}{9} \, \pi \right )^{2} + 5 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right )^{4} + 5 \, \cos \left (\frac {1}{9} \, \pi \right )^{4} \sin \left (\frac {1}{9} \, \pi \right ) - 10 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} \sin \left (\frac {1}{9} \, \pi \right )^{3} + \sin \left (\frac {1}{9} \, \pi \right )^{5} + \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} - \sqrt {3} \sin \left (\frac {1}{9} \, \pi \right )^{2} + 2 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right )\right )} \arctan \left (-\frac {{\left (-i \, \sqrt {3} - 1\right )} \cos \left (\frac {1}{9} \, \pi \right ) - 2 \, x}{-{\left (-i \, \sqrt {3} - 1\right )} \sin \left (\frac {1}{9} \, \pi \right )}\right ) - \frac {1}{18} \, {\left (5 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{4} \sin \left (\frac {4}{9} \, \pi \right ) - 10 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{2} \sin \left (\frac {4}{9} \, \pi \right )^{3} + \sqrt {3} \sin \left (\frac {4}{9} \, \pi \right )^{5} + \cos \left (\frac {4}{9} \, \pi \right )^{5} - 10 \, \cos \left (\frac {4}{9} \, \pi \right )^{3} \sin \left (\frac {4}{9} \, \pi \right )^{2} + 5 \, \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right )^{4} - 2 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right ) - \cos \left (\frac {4}{9} \, \pi \right )^{2} + \sin \left (\frac {4}{9} \, \pi \right )^{2}\right )} \log \left ({\left (-i \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right ) - \cos \left (\frac {4}{9} \, \pi \right )\right )} x + x^{2} + 1\right ) - \frac {1}{18} \, {\left (5 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{4} \sin \left (\frac {2}{9} \, \pi \right ) - 10 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{2} \sin \left (\frac {2}{9} \, \pi \right )^{3} + \sqrt {3} \sin \left (\frac {2}{9} \, \pi \right )^{5} + \cos \left (\frac {2}{9} \, \pi \right )^{5} - 10 \, \cos \left (\frac {2}{9} \, \pi \right )^{3} \sin \left (\frac {2}{9} \, \pi \right )^{2} + 5 \, \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right )^{4} - 2 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right ) - \cos \left (\frac {2}{9} \, \pi \right )^{2} + \sin \left (\frac {2}{9} \, \pi \right )^{2}\right )} \log \left ({\left (-i \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right ) - \cos \left (\frac {2}{9} \, \pi \right )\right )} x + x^{2} + 1\right ) - \frac {1}{18} \, {\left (5 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{4} \sin \left (\frac {1}{9} \, \pi \right ) - 10 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} \sin \left (\frac {1}{9} \, \pi \right )^{3} + \sqrt {3} \sin \left (\frac {1}{9} \, \pi \right )^{5} - \cos \left (\frac {1}{9} \, \pi \right )^{5} + 10 \, \cos \left (\frac {1}{9} \, \pi \right )^{3} \sin \left (\frac {1}{9} \, \pi \right )^{2} - 5 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right )^{4} + 2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \cos \left (\frac {1}{9} \, \pi \right )^{2} + \sin \left (\frac {1}{9} \, \pi \right )^{2}\right )} \log \left ({\left (i \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) + \cos \left (\frac {1}{9} \, \pi \right )\right )} x + x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.45, size = 304, normalized size = 0.81 \begin {gather*} \frac {\ln \left (x+\left (81\,x-\frac {27\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{4}\right )\,\left (-\frac {1}{162}+\frac {\sqrt {3}\,1{}\mathrm {i}}{486}\right )\right )\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}+\frac {\ln \left (x-\left (81\,x-\frac {27\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{4}\right )\,\left (\frac {1}{162}+\frac {\sqrt {3}\,1{}\mathrm {i}}{486}\right )\right )\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{1/3}\,3^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{12}+\frac {2^{1/3}\,3^{1/6}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,1{}\mathrm {i}}{4}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{1/3}\,3^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{12}-\frac {2^{1/3}\,3^{1/6}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,1{}\mathrm {i}}{4}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,3^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,3^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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