Integrand size = 29, antiderivative size = 101 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-2+7 x^3\right )}{10 x^5}+\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{-1+x^3}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {594, 597, 12, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (2 x^3-1\right )+\frac {1}{2} \log \left (-\sqrt [3]{x^3-1}-x\right )-\frac {\left (x^3-1\right )^{2/3}}{5 x^5}+\frac {7 \left (x^3-1\right )^{2/3}}{10 x^2} \]
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Rule 12
Rule 384
Rule 594
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {7-9 x^3}{x^3 \sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {7 \left (-1+x^3\right )^{2/3}}{10 x^2}+\frac {1}{10} \int -\frac {10}{\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {7 \left (-1+x^3\right )^{2/3}}{10 x^2}-\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {7 \left (-1+x^3\right )^{2/3}}{10 x^2}-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (-1+2 x^3\right )+\frac {1}{2} \log \left (-x-\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {1}{30} \left (\frac {3 \left (-1+x^3\right )^{2/3} \left (-2+7 x^3\right )}{x^5}-10 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{-1+x^3}}\right )+10 \log \left (x+\sqrt [3]{-1+x^3}\right )-5 \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
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Time = 2.61 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99
method | result | size |
pseudoelliptic | \(\frac {10 \ln \left (\frac {x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (21 x^{3}-6\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+5 x^{5} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{3}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right )-\ln \left (\frac {x^{2}-x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{30 x^{5}}\) | \(100\) |
risch | \(\frac {7 x^{6}-9 x^{3}+2}{10 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}}}-\frac {\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2 x \left (x^{3}-1\right )^{\frac {2}{3}}+2 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{2 x^{3}-1}\right )}{3}-\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2 x \left (x^{3}-1\right )^{\frac {2}{3}}+2 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{2 x^{3}-1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-x \left (x^{3}-1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{2 x^{3}-1}\right )\) | \(419\) |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (7 x^{3}-2\right )}{10 x^{5}}+32 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \ln \left (-\frac {-374952960 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+10252800 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -10252800 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-34273056 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}-421089 x \left (x^{3}-1\right )^{\frac {2}{3}}+421089 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-252248 x^{3}+2999623680 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}+51556128 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+220717}{2 x^{3}-1}\right )-\frac {\ln \left (\frac {374952960 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+10252800 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -10252800 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-26461536 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+527889 x \left (x^{3}-1\right )^{\frac {2}{3}}-527889 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-64078 x^{3}-2999623680 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-10936032 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-9154}{2 x^{3}-1}\right )}{3}-32 \ln \left (\frac {374952960 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+10252800 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -10252800 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-26461536 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+527889 x \left (x^{3}-1\right )^{\frac {2}{3}}-527889 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-64078 x^{3}-2999623680 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-10936032 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-9154}{2 x^{3}-1}\right ) \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )\) | \(488\) |
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Time = 0.53 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} - 1\right )}}{7 \, x^{3} + 1}\right ) - 5 \, x^{5} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{2 \, x^{3} - 1}\right ) - 3 \, {\left (7 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \cdot \left (3 x^{3} - 1\right )}{x^{6} \cdot \left (2 x^{3} - 1\right )}\, dx \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (3\,x^3-1\right )}{x^6\,\left (2\,x^3-1\right )} \,d x \]
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