Integrand size = 28, antiderivative size = 104 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {1}{6} \text {RootSum}\left [2-7 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-7 \text {$\#$1}+8 \text {$\#$1}^4}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(509\) vs. \(2(104)=208\).
Time = 0.36 (sec) , antiderivative size = 509, normalized size of antiderivative = 4.89, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6860, 283, 245, 399, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\frac {\left (5+\sqrt {17}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt {51}}-\frac {\left (5-\sqrt {17}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt {51}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\sqrt [3]{199+47 \sqrt {17}} \arctan \left (\frac {\frac {\sqrt [3]{7-\sqrt {17}} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {51}}+\frac {\sqrt [3]{199-47 \sqrt {17}} \arctan \left (\frac {\frac {\sqrt [3]{7+\sqrt {17}} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {51}}-\frac {\sqrt [3]{199+47 \sqrt {17}} \log \left (2 x^3-\sqrt {17}+1\right )}{24\ 2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{199-47 \sqrt {17}} \log \left (2 x^3+\sqrt {17}+1\right )}{24\ 2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{199+47 \sqrt {17}} \log \left (\frac {1}{2} \sqrt [3]{7-\sqrt {17}} x-\sqrt [3]{x^3-1}\right )}{8\ 2^{2/3} \sqrt {17}}-\frac {\sqrt [3]{199-47 \sqrt {17}} \log \left (\frac {1}{2} \sqrt [3]{7+\sqrt {17}} x-\sqrt [3]{x^3-1}\right )}{8\ 2^{2/3} \sqrt {17}}-\frac {1}{136} \left (17+5 \sqrt {17}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {1}{136} \left (17-5 \sqrt {17}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {1}{4} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{2/3}}{4 x^2} \]
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Rule 245
Rule 283
Rule 384
Rule 399
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\left (-1+x^3\right )^{2/3}}{2 x^3}+\frac {\left (-1+x^3\right )^{2/3} \left (3+x^3\right )}{2 \left (-4+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 (3+x^3\right )}{-4+x^3+x^6} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{2} \int \left (\frac {\left (1+\frac {5}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{1-\sqrt {17}+2 x^3}+\frac {\left (1-\frac {5}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{1+\sqrt {17}+2 x^3}\right ) \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\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}{34} \left (17-5 \sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{1+\sqrt {17}+2 x^3} \, dx+\frac {1}{34} \left (17+5 \sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{1-\sqrt {17}+2 x^3} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\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}{68} \left (17-5 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{34} \left (17-\sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+\sqrt {17}+2 x^3\right )} \, dx+\frac {1}{34} \left (17+\sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (1-\sqrt {17}+2 x^3\right )} \, dx+\frac {1}{68} \left (17+5 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\left (5-\sqrt {17}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {51}}+\frac {\left (5+\sqrt {17}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {51}}-\frac {\sqrt [3]{3184+752 \sqrt {17}} \arctan \left (\frac {1+\frac {\sqrt [3]{7-\sqrt {17}} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt {51}}+\frac {\sqrt [3]{3184-752 \sqrt {17}} \arctan \left (\frac {1+\frac {\sqrt [3]{7+\sqrt {17}} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt {51}}-\frac {\sqrt [3]{3184+752 \sqrt {17}} \log \left (1-\sqrt {17}+2 x^3\right )}{96 \sqrt {17}}+\frac {\sqrt [3]{3184-752 \sqrt {17}} \log \left (1+\sqrt {17}+2 x^3\right )}{96 \sqrt {17}}+\frac {\sqrt [3]{3184+752 \sqrt {17}} \log \left (\frac {1}{2} \sqrt [3]{7-\sqrt {17}} x-\sqrt [3]{-1+x^3}\right )}{32 \sqrt {17}}-\frac {\sqrt [3]{3184-752 \sqrt {17}} \log \left (\frac {1}{2} \sqrt [3]{7+\sqrt {17}} x-\sqrt [3]{-1+x^3}\right )}{32 \sqrt {17}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{136} \left (17-5 \sqrt {17}\right ) \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{136} \left (17+5 \sqrt {17}\right ) \log \left (-x+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {1}{6} \text {RootSum}\left [2-7 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-7 \text {$\#$1}+8 \text {$\#$1}^4}\&\right ] \]
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Time = 187.87 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{6}-7 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\left (\textit {\_R}^{3}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )}{8 \textit {\_R}^{4}-7 \textit {\_R}}\right ) x^{2}+3 \left (x^{3}-1\right )^{\frac {2}{3}}}{12 x^{2}}\) | \(71\) |
risch | \(\text {Expression too large to display}\) | \(6666\) |
trager | \(\text {Expression too large to display}\) | \(12143\) |
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Exception generated. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + x^{3} - 4\right )} x^{3}} \,d x } \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + x^{3} - 4\right )} x^{3}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3+2\right )}{x^3\,\left (x^6+x^3-4\right )} \,d x \]
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