Integrand size = 35, antiderivative size = 86 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2-2 x^3+x^6\right )}{x^6 \left (-4+4 x^3+x^6\right )} \, dx=\frac {\left (1-x^3\right ) \left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {1}{8} \text {RootSum}\left [-1-4 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+2 \text {$\#$1}^3}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(327\) vs. \(2(86)=172\).
Time = 0.51 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.80, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {6860, 270, 1442, 399, 245, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2-2 x^3+x^6\right )}{x^6 \left (-4+4 x^3+x^6\right )} \, dx=-\frac {\sqrt {3} \sqrt [3]{3-2 \sqrt {2}} \arctan \left (\frac {1-\frac {2^{2/3} \sqrt [3]{\sqrt {2}-1} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{16 \sqrt [6]{2}}+\frac {\sqrt {3} \sqrt [3]{3+2 \sqrt {2}} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{1+\sqrt {2}} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{16 \sqrt [6]{2}}-\frac {\sqrt [3]{3-2 \sqrt {2}} \log \left (2 x^3+4 \left (1-\sqrt {2}\right )\right )}{32 \sqrt [6]{2}}+\frac {\sqrt [3]{3+2 \sqrt {2}} \log \left (2 x^3+4 \left (1+\sqrt {2}\right )\right )}{32 \sqrt [6]{2}}+\frac {3 \sqrt [3]{3-2 \sqrt {2}} \log \left (-\sqrt [3]{x^3-1}-\sqrt [3]{\frac {1}{2} \left (\sqrt {2}-1\right )} x\right )}{32 \sqrt [6]{2}}-\frac {3 \sqrt [3]{3+2 \sqrt {2}} \log \left (\sqrt [3]{\frac {1}{2} \left (1+\sqrt {2}\right )} x-\sqrt [3]{x^3-1}\right )}{32 \sqrt [6]{2}}-\frac {\left (x^3-1\right )^{5/3}}{10 x^5} \]
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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 {3 \left (-1+x^3\right )^{2/3}}{2 \left (-4+4 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 {3}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{-4+4 x^3+x^6} \, dx \\ & = -\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {3 \int \frac {\left (-1+x^3\right )^{2/3}}{4-4 \sqrt {2}+2 x^3} \, dx}{4 \sqrt {2}}-\frac {3 \int \frac {\left (-1+x^3\right )^{2/3}}{4+4 \sqrt {2}+2 x^3} \, dx}{4 \sqrt {2}} \\ & = -\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{8} \left (3 \left (4-3 \sqrt {2}\right )\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (4-4 \sqrt {2}+2 x^3\right )} \, dx+\frac {1}{8} \left (3 \left (4+3 \sqrt {2}\right )\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (4+4 \sqrt {2}+2 x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}-\frac {\sqrt {3} \sqrt [3]{3-2 \sqrt {2}} \arctan \left (\frac {1-\frac {2^{2/3} \sqrt [3]{-1+\sqrt {2}} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt [6]{2}}+\frac {\sqrt {3} \sqrt [3]{3+2 \sqrt {2}} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{1+\sqrt {2}} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt [6]{2}}-\frac {\sqrt [3]{3-2 \sqrt {2}} \log \left (4 \left (1-\sqrt {2}\right )+2 x^3\right )}{32 \sqrt [6]{2}}+\frac {\sqrt [3]{3+2 \sqrt {2}} \log \left (4 \left (1+\sqrt {2}\right )+2 x^3\right )}{32 \sqrt [6]{2}}+\frac {3 \sqrt [3]{3-2 \sqrt {2}} \log \left (-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {2}\right )} x-\sqrt [3]{-1+x^3}\right )}{32 \sqrt [6]{2}}-\frac {3 \sqrt [3]{3+2 \sqrt {2}} \log \left (\sqrt [3]{\frac {1}{2} \left (1+\sqrt {2}\right )} x-\sqrt [3]{-1+x^3}\right )}{32 \sqrt [6]{2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2-2 x^3+x^6\right )}{x^6 \left (-4+4 x^3+x^6\right )} \, dx=-\frac {4 \left (-1+x^3\right )^{5/3}+5 x^5 \text {RootSum}\left [-1-4 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+2 \text {$\#$1}^3}\&\right ]}{40 x^5} \]
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Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(\frac {-5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{6}-4 \textit {\_Z}^{3}-1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\frac {-\textit {\_R} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{3}-1}\right ) x^{5}-4 x^{3} \left (x^{3}-1\right )^{\frac {2}{3}}+4 \left (x^{3}-1\right )^{\frac {2}{3}}}{40 x^{5}}\) | \(79\) |
risch | \(\text {Expression too large to display}\) | \(6847\) |
trager | \(\text {Expression too large to display}\) | \(11706\) |
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Exception generated. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2-2 x^3+x^6\right )}{x^6 \left (-4+4 x^3+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2-2 x^3+x^6\right )}{x^6 \left (-4+4 x^3+x^6\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.41 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2-2 x^3+x^6\right )}{x^6 \left (-4+4 x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 2 \, x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 4 \, x^{3} - 4\right )} x^{6}} \,d x } \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.41 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2-2 x^3+x^6\right )}{x^6 \left (-4+4 x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 2 \, x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 4 \, x^{3} - 4\right )} x^{6}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.41 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2-2 x^3+x^6\right )}{x^6 \left (-4+4 x^3+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6-2\,x^3+2\right )}{x^6\,\left (x^6+4\,x^3-4\right )} \,d x \]
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