Integrand size = 25, antiderivative size = 110 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{96} \text {RootSum}\left [5-8 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \]
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Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.55, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6857, 270, 283, 245, 399, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (2-i)^{2/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{2-i} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) (2+i)^{2/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{2+i} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\left (\frac {1}{48}-\frac {i}{48}\right ) \left (1+\frac {i}{2}\right )^{2/3} \log \left (-x^3+2 i\right )+\left (\frac {1}{48}+\frac {i}{48}\right ) \left (1-\frac {i}{2}\right )^{2/3} \log \left (x^3+2 i\right )-\left (\frac {1}{16}+\frac {i}{16}\right ) \left (1-\frac {i}{2}\right )^{2/3} \log \left (-\sqrt [3]{x^3-1}+\sqrt [3]{1-\frac {i}{2}} x\right )-\left (\frac {1}{16}-\frac {i}{16}\right ) \left (1+\frac {i}{2}\right )^{2/3} \log \left (-\sqrt [3]{x^3-1}+\sqrt [3]{1+\frac {i}{2}} x\right )+\frac {\left (x^3-1\right )^{5/3}}{10 x^5}-\frac {\left (x^3-1\right )^{2/3}}{8 x^2} \]
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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}}{2 x^6}+\frac {\left (-1+x^3\right )^{2/3}}{4 x^3}+\frac {\left (-2-x^3\right ) \left (-1+x^3\right )^{2/3}}{4 \left (4+x^6\right )}\right ) \, dx \\ & = \frac {1}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx+\frac {1}{4} \int \frac {\left (-2-x^3\right ) \left (-1+x^3\right )^{2/3}}{4+x^6} \, dx+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{4} \int \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (-1+x^3\right )^{2/3}}{2 i-x^3}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-1+x^3\right )^{2/3}}{2 i+x^3}\right ) \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\left (-\frac {1}{8}-\frac {i}{8}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2 i+x^3} \, dx+\left (\frac {1}{8}-\frac {i}{8}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2 i-x^3} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\left (-\frac {1}{8}-\frac {i}{8}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\left (-\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\left (-\frac {1}{8}+\frac {3 i}{8}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (2 i+x^3\right )} \, dx+\left (\frac {1}{8}+\frac {3 i}{8}\right ) \int \frac {1}{\left (2 i-x^3\right ) \sqrt [3]{-1+x^3}} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (2-i)^{2/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{2-i} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) (2+i)^{2/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{2+i} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\left (\frac {1}{48}-\frac {i}{48}\right ) \left (1+\frac {i}{2}\right )^{2/3} \log \left (2 i-x^3\right )+\left (\frac {1}{48}+\frac {i}{48}\right ) \left (1-\frac {i}{2}\right )^{2/3} \log \left (2 i+x^3\right )-\left (\frac {1}{16}+\frac {i}{16}\right ) \left (1-\frac {i}{2}\right )^{2/3} \log \left (\sqrt [3]{1-\frac {i}{2}} x-\sqrt [3]{-1+x^3}\right )-\left (\frac {1}{16}-\frac {i}{16}\right ) \left (1+\frac {i}{2}\right )^{2/3} \log \left (\sqrt [3]{1+\frac {i}{2}} x-\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{96} \text {RootSum}\left [5-8 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \]
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Time = 196.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.75
method | result | size |
pseudoelliptic | \(\frac {-5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{6}-8 \textit {\_Z}^{3}+5\right )}{\sum }\frac {\left (6 \textit {\_R}^{3}-5\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{4}-\textit {\_R}}\right ) x^{5}-12 x^{3} \left (x^{3}-1\right )^{\frac {2}{3}}-48 \left (x^{3}-1\right )^{\frac {2}{3}}}{480 x^{5}}\) | \(83\) |
risch | \(\text {Expression too large to display}\) | \(6547\) |
trager | \(\text {Expression too large to display}\) | \(8788\) |
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Exception generated. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 6.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.25 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} + 2\right )}{x^{6} \left (x^{6} + 4\right )}\, dx \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 4\right )} x^{6}} \,d x } \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 4\right )} x^{6}} \,d x } \]
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Not integrable
Time = 5.94 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3+2\right )}{x^6\,\left (x^6+4\right )} \,d x \]
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