Integrand size = 30, antiderivative size = 86 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-4-2 x^3+x^6\right )} \, dx=\frac {\left (1-x^3\right ) \left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {1}{12} \text {RootSum}\left [5-10 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-5+4 \text {$\#$1}^3}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(319\) vs. \(2(86)=172\).
Time = 0.45 (sec) , antiderivative size = 319, normalized size of antiderivative = 3.71, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6860, 270, 1442, 399, 245, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-4-2 x^3+x^6\right )} \, dx=-\frac {\sqrt [3]{\frac {1}{5} \left (3 \sqrt {5}-5\right )} \arctan \left (\frac {\frac {\sqrt [3]{2 \left (5-\sqrt {5}\right )} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{8 \sqrt {3}}+\frac {\sqrt [3]{\frac {1}{5} \left (5+3 \sqrt {5}\right )} \arctan \left (\frac {\frac {\sqrt [3]{2 \left (5+\sqrt {5}\right )} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{8 \sqrt {3}}+\frac {1}{48} \sqrt [3]{\frac {1}{5} \left (5+3 \sqrt {5}\right )} \log \left (2 x^3-2 \left (1-\sqrt {5}\right )\right )-\frac {1}{48} \sqrt [3]{\frac {1}{5} \left (3 \sqrt {5}-5\right )} \log \left (2 x^3-2 \left (1+\sqrt {5}\right )\right )+\frac {1}{16} \sqrt [3]{\frac {1}{5} \left (3 \sqrt {5}-5\right )} \log \left (\frac {\sqrt [3]{5-\sqrt {5}} x}{2^{2/3}}-\sqrt [3]{x^3-1}\right )-\frac {1}{16} \sqrt [3]{\frac {1}{5} \left (5+3 \sqrt {5}\right )} \log \left (\frac {\sqrt [3]{5+\sqrt {5}} x}{2^{2/3}}-\sqrt [3]{x^3-1}\right )-\frac {\left (x^3-1\right )^{5/3}}{10 x^5} \]
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Rule 245
Rule 270
Rule 384
Rule 399
Rule 1442
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\left (-1+x^3\right )^{2/3}}{2 x^6}+\frac {\left (-1+x^3\right )^{2/3}}{2 \left (-4-2 x^3+x^6\right )}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\right )+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{-4-2 x^3+x^6} \, dx \\ & = -\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {\int \frac {\left (-1+x^3\right )^{2/3}}{-2-2 \sqrt {5}+2 x^3} \, dx}{2 \sqrt {5}}-\frac {\int \frac {\left (-1+x^3\right )^{2/3}}{-2+2 \sqrt {5}+2 x^3} \, dx}{2 \sqrt {5}} \\ & = -\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3} \left (-2-2 \sqrt {5}+2 x^3\right )} \, dx+\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3} \left (-2+2 \sqrt {5}+2 x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}-\frac {\sqrt [3]{\frac {1}{5} \left (-5+3 \sqrt {5}\right )} \arctan \left (\frac {1+\frac {\sqrt [3]{2 \left (5-\sqrt {5}\right )} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{8 \sqrt {3}}+\frac {\sqrt [3]{\frac {1}{5} \left (5+3 \sqrt {5}\right )} \arctan \left (\frac {1+\frac {\sqrt [3]{2 \left (5+\sqrt {5}\right )} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{8 \sqrt {3}}+\frac {1}{48} \sqrt [3]{\frac {1}{5} \left (5+3 \sqrt {5}\right )} \log \left (-2 \left (1-\sqrt {5}\right )+2 x^3\right )-\frac {1}{48} \sqrt [3]{\frac {1}{5} \left (-5+3 \sqrt {5}\right )} \log \left (-2 \left (1+\sqrt {5}\right )+2 x^3\right )+\frac {1}{16} \sqrt [3]{\frac {1}{5} \left (-5+3 \sqrt {5}\right )} \log \left (\frac {\sqrt [3]{5-\sqrt {5}} x}{2^{2/3}}-\sqrt [3]{-1+x^3}\right )-\frac {1}{16} \sqrt [3]{\frac {1}{5} \left (5+3 \sqrt {5}\right )} \log \left (\frac {\sqrt [3]{5+\sqrt {5}} x}{2^{2/3}}-\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-4-2 x^3+x^6\right )} \, dx=-\frac {6 \left (-1+x^3\right )^{5/3}+5 x^5 \text {RootSum}\left [5-10 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-5+4 \text {$\#$1}^3}\&\right ]}{60 x^5} \]
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Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(\frac {-5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{6}-10 \textit {\_Z}^{3}+5\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\frac {-\textit {\_R} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )}{4 \textit {\_R}^{3}-5}\right ) x^{5}-6 x^{3} \left (x^{3}-1\right )^{\frac {2}{3}}+6 \left (x^{3}-1\right )^{\frac {2}{3}}}{60 x^{5}}\) | \(79\) |
risch | \(\text {Expression too large to display}\) | \(6931\) |
trager | \(\text {Expression too large to display}\) | \(10910\) |
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Exception generated. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-4-2 x^3+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-4-2 x^3+x^6\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.35 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-4-2 x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 2 \, x^{3} - 4\right )} x^{6}} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.35 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-4-2 x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 2 \, x^{3} - 4\right )} x^{6}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.38 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-4-2 x^3+x^6\right )} \, dx=\int -\frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3+2\right )}{x^6\,\left (-x^6+2\,x^3+4\right )} \,d x \]
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