Integrand size = 30, antiderivative size = 112 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-4+9 x^3\right )}{20 x^5}-\frac {1}{12} \text {RootSum}\left [7-10 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-7 \log (x)+7 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \]
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Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 679, normalized size of antiderivative = 6.06, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6860, 270, 283, 245, 399, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\frac {1}{12} \left (\sqrt {3}+i\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\frac {1}{12} \left (-\sqrt {3}+i\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\left (\sqrt {3}+i\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}} \sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{12 \left (\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}\right )^{2/3}}+\frac {\left (-\sqrt {3}+i\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+2 i}} \sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{12 \left (\frac {\sqrt {3}+i}{\sqrt {3}+2 i}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (2 x^3+2 \left (1-i \sqrt {3}\right )\right )}{72 \left (\frac {\sqrt {3}+i}{\sqrt {3}+2 i}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (2 x^3+2 \left (1+i \sqrt {3}\right )\right )}{72 \left (\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (-\sqrt [3]{x^3-1}+\frac {x}{\sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}}}\right )}{24 \left (\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}\right )^{2/3}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (-\sqrt [3]{x^3-1}+\frac {x}{\sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+2 i}}}\right )}{24 \left (\frac {\sqrt {3}+i}{\sqrt {3}+2 i}\right )^{2/3}}-\frac {1}{24} \left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {1}{24} \left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {1}{4} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{4 x^2} \]
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Rule 245
Rule 270
Rule 283
Rule 384
Rule 399
Rule 6860
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}}{2 x^3}+\frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{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^3} \, dx\right )+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{4+2 x^3+x^6} \, dx+\int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{2} \int \left (\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \left (-1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3}+\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \left (-1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3}\right ) \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 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 )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3} \, dx+\frac {1}{6} \left (3+i \sqrt {3}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 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 )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{12} \left (3-i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{12} \left (3+i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{6} \left (-3+5 i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (2-2 i \sqrt {3}+2 x^3\right )} \, dx-\frac {1}{6} \left (3+5 i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (2+2 i \sqrt {3}+2 x^3\right )} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 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 )}{2 \sqrt {3}}-\frac {1}{12} \left (i-\sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {1}{12} \left (i+\sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-\frac {\left (5-i \sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {i-\sqrt {3}}{2 i-\sqrt {3}}} \sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{12 \left (-i+\sqrt {3}\right )^{2/3} \sqrt [3]{-2 i+\sqrt {3}}}-\frac {\left (5+i \sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {i+\sqrt {3}}{2 i+\sqrt {3}}} \sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{12 \left (i+\sqrt {3}\right )^{2/3} \sqrt [3]{2 i+\sqrt {3}}}-\frac {\left (3 i+5 \sqrt {3}\right ) \log \left (2 \left (1-i \sqrt {3}\right )+2 x^3\right )}{72 \left (i+\sqrt {3}\right )^{2/3} \sqrt [3]{2 i+\sqrt {3}}}+\frac {\left (3 i-5 \sqrt {3}\right ) \log \left (2 \left (1+i \sqrt {3}\right )+2 x^3\right )}{72 \left (-i+\sqrt {3}\right )^{2/3} \sqrt [3]{-2 i+\sqrt {3}}}-\frac {\left (3 i-5 \sqrt {3}\right ) \log \left (\frac {x}{\sqrt [3]{\frac {i-\sqrt {3}}{2 i-\sqrt {3}}}}-\sqrt [3]{-1+x^3}\right )}{24 \left (-i+\sqrt {3}\right )^{2/3} \sqrt [3]{-2 i+\sqrt {3}}}+\frac {\left (3 i+5 \sqrt {3}\right ) \log \left (\frac {x}{\sqrt [3]{\frac {i+\sqrt {3}}{2 i+\sqrt {3}}}}-\sqrt [3]{-1+x^3}\right )}{24 \left (i+\sqrt {3}\right )^{2/3} \sqrt [3]{2 i+\sqrt {3}}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{24} \left (3-i \sqrt {3}\right ) \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{24} \left (3+i \sqrt {3}\right ) \log \left (-x+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-4+9 x^3\right )}{20 x^5}-\frac {1}{12} \text {RootSum}\left [7-10 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-7 \log (x)+7 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \]
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Timed out.
\[\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (x^{6}+4\right )}{x^{6} \left (x^{6}+2 x^{3}+4\right )}d x\]
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Exception generated. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\right )} x^{6}} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\right )} x^{6}} \,d x } \]
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Not integrable
Time = 6.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+4\right )}{x^6\,\left (x^6+2\,x^3+4\right )} \,d x \]
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