Integrand size = 30, antiderivative size = 100 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {1}{3} \text {RootSum}\left [-1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(505\) vs. \(2(100)=200\).
Time = 0.54 (sec) , antiderivative size = 505, normalized size of antiderivative = 5.05, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6860, 283, 245, 399, 384} \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=-\frac {\left (3+\sqrt {5}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {15}}+\frac {\left (3-\sqrt {5}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {15}}+\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} \arctan \left (\frac {1-\frac {2^{2/3} \sqrt [3]{\sqrt {5}-1} x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{\sqrt {15}}-\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{1+\sqrt {5}} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {15}}-\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} \log \left (2 x^3-\sqrt {5}-1\right )}{6 \sqrt {5}}+\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} \log \left (2 x^3+\sqrt {5}-1\right )}{6 \sqrt {5}}-\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} \log \left (-\sqrt [3]{x^3+1}-\sqrt [3]{\frac {1}{2} \left (\sqrt {5}-1\right )} x\right )}{2 \sqrt {5}}+\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} \log \left (\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x-\sqrt [3]{x^3+1}\right )}{2 \sqrt {5}}+\frac {1}{20} \left (5+3 \sqrt {5}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{20} \left (5-3 \sqrt {5}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \]
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Rule 245
Rule 283
Rule 384
Rule 399
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {\left (2-x^3\right ) \left (1+x^3\right )^{2/3}}{-1-x^3+x^6}\right ) \, dx \\ & = \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {\left (2-x^3\right ) \left (1+x^3\right )^{2/3}}{-1-x^3+x^6} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\int \left (\frac {\left (-1+\frac {3}{\sqrt {5}}\right ) \left (1+x^3\right )^{2/3}}{-1-\sqrt {5}+2 x^3}+\frac {\left (-1-\frac {3}{\sqrt {5}}\right ) \left (1+x^3\right )^{2/3}}{-1+\sqrt {5}+2 x^3}\right ) \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{5} \left (-5+3 \sqrt {5}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-1-\sqrt {5}+2 x^3} \, dx-\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-1+\sqrt {5}+2 x^3} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {2 \int \frac {1}{\sqrt [3]{1+x^3} \left (-1-\sqrt {5}+2 x^3\right )} \, dx}{\sqrt {5}}-\frac {2 \int \frac {1}{\sqrt [3]{1+x^3} \left (-1+\sqrt {5}+2 x^3\right )} \, dx}{\sqrt {5}}+\frac {1}{10} \left (-5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt [3]{1+x^3}} \, dx-\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt [3]{1+x^3}} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (3-\sqrt {5}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {15}}-\frac {\left (3+\sqrt {5}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {15}}+\frac {2 \sqrt [3]{2} \arctan \left (\frac {1-\frac {2^{2/3} \sqrt [3]{-1+\sqrt {5}} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {15} \left (-1+\sqrt {5}\right )^{4/3}}-\frac {2 \sqrt [3]{2} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{1+\sqrt {5}} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {15} \left (1+\sqrt {5}\right )^{4/3}}-\frac {\sqrt [3]{2} \log \left (-1-\sqrt {5}+2 x^3\right )}{3 \sqrt {5} \left (1+\sqrt {5}\right )^{4/3}}+\frac {\sqrt [3]{2} \log \left (-1+\sqrt {5}+2 x^3\right )}{3 \sqrt {5} \left (-1+\sqrt {5}\right )^{4/3}}-\frac {\sqrt [3]{2} \log \left (-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x-\sqrt [3]{1+x^3}\right )}{\sqrt {5} \left (-1+\sqrt {5}\right )^{4/3}}+\frac {\sqrt [3]{2} \log \left (\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x-\sqrt [3]{1+x^3}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )^{4/3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{20} \left (5-3 \sqrt {5}\right ) \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{20} \left (5+3 \sqrt {5}\right ) \log \left (-x+\sqrt [3]{1+x^3}\right ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {1}{3} \text {RootSum}\left [-1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]
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Time = 83.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}-1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{4}-\textit {\_R}}\right ) x^{2}-3 \left (x^{3}+1\right )^{\frac {2}{3}}}{6 x^{2}}\) | \(69\) |
risch | \(\text {Expression too large to display}\) | \(9436\) |
trager | \(\text {Expression too large to display}\) | \(14259\) |
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Exception generated. \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.30 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{6} - x^{3} - 1\right )} x^{3}} \,d x } \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.30 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{6} - x^{3} - 1\right )} x^{3}} \,d x } \]
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Not integrable
Time = 6.55 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.32 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=-\int \frac {\left (x^3-1\right )\,{\left (x^3+1\right )}^{2/3}}{x^3\,\left (-x^6+x^3+1\right )} \,d x \]
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