Integrand size = 35, antiderivative size = 86 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4-2 x^3+x^6\right )}{x^6 \left (-8+4 x^3+x^6\right )} \, dx=\frac {\left (1-x^3\right ) \left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {1}{8} \text {RootSum}\left [3-12 \text {$\#$1}^3+8 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-3+4 \text {$\#$1}^3}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(313\) vs. \(2(86)=172\).
Time = 0.77 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.64, 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} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4-2 x^3+x^6\right )}{x^6 \left (-8+4 x^3+x^6\right )} \, dx=-\frac {1}{16} \sqrt [6]{3} \sqrt [3]{2 \sqrt {3}-3} \arctan \left (\frac {\frac {\sqrt [3]{2 \left (3-\sqrt {3}\right )} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )+\frac {1}{16} \sqrt [6]{3} \sqrt [3]{3+2 \sqrt {3}} \arctan \left (\frac {\frac {\sqrt [3]{2 \left (3+\sqrt {3}\right )} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\frac {1}{32} \sqrt [3]{\frac {1}{3} \left (2 \sqrt {3}-3\right )} \log \left (2 x^3+4 \left (1-\sqrt {3}\right )\right )+\frac {1}{32} \sqrt [3]{\frac {1}{3} \left (3+2 \sqrt {3}\right )} \log \left (2 x^3+4 \left (1+\sqrt {3}\right )\right )+\frac {1}{32} 3^{2/3} \sqrt [3]{2 \sqrt {3}-3} \log \left (\frac {\sqrt [3]{3-\sqrt {3}} x}{2^{2/3}}-\sqrt [3]{x^3-1}\right )-\frac {1}{32} 3^{2/3} \sqrt [3]{3+2 \sqrt {3}} \log \left (\frac {\sqrt [3]{3+\sqrt {3}} x}{2^{2/3}}-\sqrt [3]{x^3-1}\right )-\frac {\left (x^3-1\right )^{5/3}}{10 x^5} \]
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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 {3 \left (-1+x^3\right )^{2/3}}{2 \left (-8+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}}{-8+4 x^3+x^6} \, dx \\ & = -\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \sqrt {3} \int \frac {\left (-1+x^3\right )^{2/3}}{4-4 \sqrt {3}+2 x^3} \, dx-\frac {1}{4} \sqrt {3} \int \frac {\left (-1+x^3\right )^{2/3}}{4+4 \sqrt {3}+2 x^3} \, dx \\ & = -\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \left (3 \left (2-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (4-4 \sqrt {3}+2 x^3\right )} \, dx+\frac {1}{4} \left (3 \left (2+\sqrt {3}\right )\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (4+4 \sqrt {3}+2 x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}-\frac {1}{16} \sqrt [6]{3} \sqrt [3]{-3+2 \sqrt {3}} \arctan \left (\frac {1+\frac {\sqrt [3]{2 \left (3-\sqrt {3}\right )} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {1}{16} \sqrt [6]{3} \sqrt [3]{3+2 \sqrt {3}} \arctan \left (\frac {1+\frac {\sqrt [3]{2 \left (3+\sqrt {3}\right )} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-\frac {1}{32} \sqrt [3]{-1+\frac {2}{\sqrt {3}}} \log \left (4 \left (1-\sqrt {3}\right )+2 x^3\right )+\frac {1}{32} \sqrt [3]{1+\frac {2}{\sqrt {3}}} \log \left (4 \left (1+\sqrt {3}\right )+2 x^3\right )+\frac {3}{32} \sqrt [3]{-1+\frac {2}{\sqrt {3}}} \log \left (\frac {\sqrt [3]{3-\sqrt {3}} x}{2^{2/3}}-\sqrt [3]{-1+x^3}\right )-\frac {3}{32} \sqrt [3]{1+\frac {2}{\sqrt {3}}} \log \left (\frac {\sqrt [3]{3+\sqrt {3}} x}{2^{2/3}}-\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4-2 x^3+x^6\right )}{x^6 \left (-8+4 x^3+x^6\right )} \, dx=-\frac {4 \left (-1+x^3\right )^{5/3}+5 x^5 \text {RootSum}\left [3-12 \text {$\#$1}^3+8 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-3+4 \text {$\#$1}^3}\&\right ]}{40 x^5} \]
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Time = 290.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(\frac {-5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{6}-12 \textit {\_Z}^{3}+3\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\frac {-\textit {\_R} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )}{4 \textit {\_R}^{3}-3}\right ) x^{5}-4 x^{3} \left (x^{3}-1\right )^{\frac {2}{3}}+4 \left (x^{3}-1\right )^{\frac {2}{3}}}{40 x^{5}}\) | \(79\) |
risch | \(\text {Expression too large to display}\) | \(8349\) |
trager | \(\text {Expression too large to display}\) | \(8761\) |
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Exception generated. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4-2 x^3+x^6\right )}{x^6 \left (-8+4 x^3+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4-2 x^3+x^6\right )}{x^6 \left (-8+4 x^3+x^6\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.41 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4-2 x^3+x^6\right )}{x^6 \left (-8+4 x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 2 \, x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 4 \, x^{3} - 8\right )} x^{6}} \,d x } \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.41 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4-2 x^3+x^6\right )}{x^6 \left (-8+4 x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 2 \, x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 4 \, x^{3} - 8\right )} x^{6}} \,d x } \]
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Not integrable
Time = 5.79 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.41 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4-2 x^3+x^6\right )}{x^6 \left (-8+4 x^3+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6-2\,x^3+4\right )}{x^6\,\left (x^6+4\,x^3-8\right )} \,d x \]
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