Optimal. Leaf size=416 \[ \frac {\left (3+i \sqrt {3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}-\frac {1}{x}-\frac {\left (3+i \sqrt {3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}-\frac {\left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (-\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}} \]
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Rubi [A] time = 0.28, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {1504, 1374, 292, 31, 634, 617, 204, 628} \[ \frac {\left (3+i \sqrt {3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}-\frac {1}{x}-\frac {\left (3+i \sqrt {3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}-\frac {\left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (-\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 617
Rule 628
Rule 634
Rule 1374
Rule 1504
Rubi steps
\begin {align*} \int \frac {1-x^3}{x^2 \left (1-x^3+x^6\right )} \, dx &=-\frac {1}{x}-\int \frac {x^4}{1-x^3+x^6} \, dx\\ &=-\frac {1}{x}+\frac {1}{6} \left (-3+i \sqrt {3}\right ) \int \frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}+x^3} \, dx-\frac {1}{6} \left (3+i \sqrt {3}\right ) \int \frac {x}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}+x^3} \, dx\\ &=-\frac {1}{x}+-\frac {\left (-3-i \sqrt {3}\right ) \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}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3-i \sqrt {3}\right ) \int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}+x} \, dx}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \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}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}-\frac {\left (3+i \sqrt {3}\right ) \int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+x} \, dx}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}\\ &=-\frac {1}{x}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+-\frac {\left (-3-i \sqrt {3}\right ) \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}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {1}{12} \left (-3+i \sqrt {3}\right ) \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+\frac {\left (3-i \sqrt {3}\right ) \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}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}-\frac {1}{12} \left (3+i \sqrt {3}\right ) \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\\ &=-\frac {1}{x}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+\frac {\left (3+i \sqrt {3}\right ) \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 )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \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 )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+-\frac {\left (-3-i \sqrt {3}\right ) \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 )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \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 )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\\ &=-\frac {1}{x}-\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+\frac {\left (3+i \sqrt {3}\right ) \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 )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \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 )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 47, normalized size = 0.11 \[ -\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\text {$\#$1}^2 \log (x-\text {$\#$1})}{2 \text {$\#$1}^3-1}\& \right ]-\frac {1}{x} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.32, size = 1598, normalized size = 3.84 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.71, size = 829, normalized size = 1.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 46, normalized size = 0.11 \[ -\frac {\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )^{4} \ln \left (-\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )+x \right )}{3 \left (2 \RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )^{5}-\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )^{2}\right )}-\frac {1}{x} \]
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 \[ -\frac {1}{x} - \int \frac {x^{4}}{x^{6} - x^{3} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 313, normalized size = 0.75 \[ \frac {\ln \left (-x+\left (162\,x+\frac {27\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{4}\right )\,\left (\frac {1}{162}+\frac {\sqrt {3}\,1{}\mathrm {i}}{486}\right )\right )\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}+\frac {\ln \left (-x-\left (162\,x+\frac {27\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{4}\right )\,\left (-\frac {1}{162}+\frac {\sqrt {3}\,1{}\mathrm {i}}{486}\right )\right )\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}-\frac {1}{x}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{1/3}\,3^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{12}-\frac {2^{1/3}\,3^{1/6}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,1{}\mathrm {i}}{4}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{1/3}\,3^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{12}+\frac {2^{1/3}\,3^{1/6}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,1{}\mathrm {i}}{4}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,3^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,3^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 31, normalized size = 0.07 \[ - \operatorname {RootSum} {\left (19683 t^{6} + 243 t^{3} + 1, \left (t \mapsto t \log {\left (6561 t^{5} + 54 t^{2} + x \right )} \right )\right )} - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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