Integrand size = 30, antiderivative size = 103 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-2+x^3+2 x^6\right )} \, dx=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{3} \text {RootSum}\left [1-5 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-2 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(327\) vs. \(2(103)=206\).
Time = 0.51 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.17, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6860, 283, 245, 1532, 384} \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-2+x^3+2 x^6\right )} \, dx=\frac {\sqrt [3]{\frac {1}{2} \left (5 \sqrt {17}-19\right )} \arctan \left (\frac {\frac {\sqrt [3]{2 \left (5-\sqrt {17}\right )} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \arctan \left (\frac {\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (4 x^3-\sqrt {17}+1\right )}{6 \sqrt {17}}+\frac {\sqrt [3]{\frac {1}{2} \left (5 \sqrt {17}-19\right )} \log \left (4 x^3+\sqrt {17}+1\right )}{6 \sqrt {17}}-\frac {\sqrt [3]{\frac {1}{2} \left (5 \sqrt {17}-19\right )} \log \left (\frac {\sqrt [3]{5-\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{x^3+1}\right )}{2 \sqrt {17}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\frac {\sqrt [3]{5+\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{x^3+1}\right )}{2 \sqrt {17}}-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \]
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Rule 245
Rule 283
Rule 384
Rule 1532
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (1+x^3\right )^{2/3}}{x^3}-\frac {2 x^3 \left (1+x^3\right )^{2/3}}{-2+x^3+2 x^6}\right ) \, dx \\ & = -\left (2 \int \frac {x^3 \left (1+x^3\right )^{2/3}}{-2+x^3+2 x^6} \, dx\right )+\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \frac {-2-x^3}{\sqrt [3]{1+x^3} \left (-2+x^3+2 x^6\right )} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \left (\frac {-1-\frac {7}{\sqrt {17}}}{\sqrt [3]{1+x^3} \left (1-\sqrt {17}+4 x^3\right )}+\frac {-1+\frac {7}{\sqrt {17}}}{\sqrt [3]{1+x^3} \left (1+\sqrt {17}+4 x^3\right )}\right ) \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{17} \left (-17+7 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (1+\sqrt {17}+4 x^3\right )} \, dx-\frac {1}{17} \left (17+7 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (1-\sqrt {17}+4 x^3\right )} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\sqrt [3]{\frac {1}{2} \left (-19+5 \sqrt {17}\right )} \arctan \left (\frac {1+\frac {\sqrt [3]{2 \left (5-\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \arctan \left (\frac {1+\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (1-\sqrt {17}+4 x^3\right )}{6 \sqrt {17}}+\frac {\sqrt [3]{\frac {1}{2} \left (-19+5 \sqrt {17}\right )} \log \left (1+\sqrt {17}+4 x^3\right )}{6 \sqrt {17}}-\frac {\sqrt [3]{\frac {1}{2} \left (-19+5 \sqrt {17}\right )} \log \left (\frac {\sqrt [3]{5-\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{1+x^3}\right )}{2 \sqrt {17}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\frac {\sqrt [3]{5+\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{1+x^3}\right )}{2 \sqrt {17}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-2+x^3+2 x^6\right )} \, dx=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{3} \text {RootSum}\left [1-5 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-2 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \]
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Time = 151.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(\frac {-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-5 \textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}-1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{4 \textit {\_R}^{4}-5 \textit {\_R}}\right ) x^{2}-3 \left (x^{3}+1\right )^{\frac {2}{3}}}{6 x^{2}}\) | \(73\) |
risch | \(\text {Expression too large to display}\) | \(7989\) |
trager | \(\text {Expression too large to display}\) | \(12165\) |
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Exception generated. \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-2+x^3+2 x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-2+x^3+2 x^6\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-2+x^3+2 x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{6} + x^{3} - 2\right )} x^{3}} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-2+x^3+2 x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{6} + x^{3} - 2\right )} x^{3}} \,d x } \]
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Not integrable
Time = 6.39 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-2+x^3+2 x^6\right )} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^3-2\right )}{x^3\,\left (2\,x^6+x^3-2\right )} \,d x \]
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