Integrand size = 32, antiderivative size = 106 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+x^3+2 x^6\right )}{x^6 \left (-1+2 x^6\right )} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (2+7 x^3\right )}{10 x^5}-\frac {1}{6} \text {RootSum}\left [-1-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x)-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+5 \log (x) \text {$\#$1}^3-5 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(465\) vs. \(2(106)=212\).
Time = 0.47 (sec) , antiderivative size = 465, normalized size of antiderivative = 4.39, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6857, 270, 283, 245, 399, 384} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+x^3+2 x^6\right )}{x^6 \left (-1+2 x^6\right )} \, dx=\frac {\left (1+2 \sqrt {2}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\left (1-2 \sqrt {2}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\sqrt [3]{116 \sqrt {2}-163} \arctan \left (\frac {1-\frac {2 \sqrt [3]{\sqrt {2}-1} x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\sqrt [3]{163+116 \sqrt {2}} \arctan \left (\frac {\frac {2 \sqrt [3]{1+\sqrt {2}} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{12} \sqrt [3]{163+116 \sqrt {2}} \log \left (1-\sqrt {2} x^3\right )+\frac {1}{12} \sqrt [3]{116 \sqrt {2}-163} \log \left (\sqrt {2} x^3+1\right )-\frac {1}{4} \sqrt [3]{116 \sqrt {2}-163} \log \left (-\sqrt [3]{x^3+1}-\sqrt [3]{\sqrt {2}-1} x\right )+\frac {1}{4} \sqrt [3]{163+116 \sqrt {2}} \log \left (\sqrt [3]{1+\sqrt {2}} x-\sqrt [3]{x^3+1}\right )-\frac {1}{4} \left (1+2 \sqrt {2}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {1}{4} \left (1-2 \sqrt {2}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {\left (x^3+1\right )^{5/3}}{5 x^5}+\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \]
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Rule 245
Rule 270
Rule 283
Rule 384
Rule 399
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\left (1+x^3\right )^{2/3}}{x^6}-\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {2 \left (1+x^3\right )^{2/3} \left (2+x^3\right )}{-1+2 x^6}\right ) \, dx \\ & = 2 \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{-1+2 x^6} \, dx-\int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx-\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx \\ & = \frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+2 \int \left (-\frac {\left (1+2 \sqrt {2}\right ) \left (1+x^3\right )^{2/3}}{2 \sqrt {2} \left (1-\sqrt {2} x^3\right )}+\frac {\left (1-2 \sqrt {2}\right ) \left (1+x^3\right )^{2/3}}{2 \sqrt {2} \left (1+\sqrt {2} x^3\right )}\right ) \, dx-\int \frac {1}{\sqrt [3]{1+x^3}} \, dx \\ & = \frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\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}{2} \left (-4+\sqrt {2}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1+\sqrt {2} x^3} \, dx-\frac {1}{2} \left (4+\sqrt {2}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1-\sqrt {2} x^3} \, dx \\ & = \frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\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}{2} \left (1-2 \sqrt {2}\right ) \int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\frac {1}{2} \left (1+2 \sqrt {2}\right ) \int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\frac {1}{2} \left (-5+3 \sqrt {2}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (1+\sqrt {2} x^3\right )} \, dx-\frac {1}{2} \left (5+3 \sqrt {2}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (1-\sqrt {2} x^3\right )} \, dx \\ & = \frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (1-2 \sqrt {2}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\left (1+2 \sqrt {2}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\sqrt [3]{-163+116 \sqrt {2}} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-1+\sqrt {2}} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\sqrt [3]{163+116 \sqrt {2}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{1+\sqrt {2}} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{12} \sqrt [3]{163+116 \sqrt {2}} \log \left (1-\sqrt {2} x^3\right )+\frac {1}{12} \sqrt [3]{-163+116 \sqrt {2}} \log \left (1+\sqrt {2} x^3\right )-\frac {1}{4} \sqrt [3]{-163+116 \sqrt {2}} \log \left (-\sqrt [3]{-1+\sqrt {2}} x-\sqrt [3]{1+x^3}\right )+\frac {1}{4} \sqrt [3]{163+116 \sqrt {2}} \log \left (\sqrt [3]{1+\sqrt {2}} x-\sqrt [3]{1+x^3}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {1}{4} \left (1-2 \sqrt {2}\right ) \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {1}{4} \left (1+2 \sqrt {2}\right ) \log \left (-x+\sqrt [3]{1+x^3}\right ) \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+x^3+2 x^6\right )}{x^6 \left (-1+2 x^6\right )} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (2+7 x^3\right )}{10 x^5}-\frac {1}{6} \text {RootSum}\left [-1-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x)-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+5 \log (x) \text {$\#$1}^3-5 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \]
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Time = 108.38 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 \textit {\_Z}^{3}-1\right )}{\sum }\frac {\left (5 \textit {\_R}^{3}+1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{4}-\textit {\_R}}\right ) x^{5}+21 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}+6 \left (x^{3}+1\right )^{\frac {2}{3}}}{30 x^{5}}\) | \(81\) |
risch | \(\text {Expression too large to display}\) | \(8584\) |
trager | \(\text {Expression too large to display}\) | \(15409\) |
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Exception generated. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+x^3+2 x^6\right )}{x^6 \left (-1+2 x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 23.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.32 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+x^3+2 x^6\right )}{x^6 \left (-1+2 x^6\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \cdot \left (2 x^{6} + x^{3} + 1\right )}{x^{6} \cdot \left (2 x^{6} - 1\right )}\, dx \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.30 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+x^3+2 x^6\right )}{x^6 \left (-1+2 x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} - 1\right )} x^{6}} \,d x } \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.30 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+x^3+2 x^6\right )}{x^6 \left (-1+2 x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} - 1\right )} x^{6}} \,d x } \]
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Not integrable
Time = 6.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.30 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+x^3+2 x^6\right )}{x^6 \left (-1+2 x^6\right )} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (2\,x^6+x^3+1\right )}{x^6\,\left (2\,x^6-1\right )} \,d x \]
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