Optimal. Leaf size=375 \[ -\frac {i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+i \sqrt {3}}}+\frac {i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+\frac {i \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}} \]
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Rubi [A] time = 0.25, 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 = {1375, 292, 31, 634, 617, 204, 628} \[ -\frac {i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+i \sqrt {3}}}+\frac {i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+\frac {i \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 617
Rule 628
Rule 634
Rule 1375
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 \operatorname {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 \operatorname {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] time = 0.01, size = 40, normalized size = 0.11 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\log (x-\text {$\#$1})}{2 \text {$\#$1}^4-\text {$\#$1}}\& \right ] \]
Antiderivative was successfully verified.
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fricas [B] time = 1.32, size = 1583, normalized size = 4.22 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.57, size = 812, normalized size = 2.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 38, normalized size = 0.10 \[ \frac {\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )+x \right )}{6 \RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )^{5}-3 \RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )^{2}} \]
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 \[ \int \frac {x}{x^{6} - x^{3} + 1}\,{d x} \]
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 \[ \frac {\ln \left (x+\left (81\,x-\frac {27\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{4}\right )\,\left (-\frac {1}{162}+\frac {\sqrt {3}\,1{}\mathrm {i}}{486}\right )\right )\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}+\frac {\ln \left (x-\left (81\,x-\frac {27\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{4}\right )\,\left (\frac {1}{162}+\frac {\sqrt {3}\,1{}\mathrm {i}}{486}\right )\right )\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{1/3}\,3^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{12}+\frac {2^{1/3}\,3^{1/6}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,1{}\mathrm {i}}{4}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{1/3}\,3^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{12}-\frac {2^{1/3}\,3^{1/6}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,1{}\mathrm {i}}{4}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,3^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,3^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 26, normalized size = 0.07 \[ \operatorname {RootSum} {\left (19683 t^{6} - 243 t^{3} + 1, \left (t \mapsto t \log {\left (6561 t^{5} - 27 t^{2} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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