Integrand size = 21, antiderivative size = 89 \[ \int \frac {-3+x}{\sqrt [3]{-1+x^2} \left (2+x+x^2\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{2+2 x+\sqrt [3]{-1+x^2}}\right )+\log \left (-1-x+\sqrt [3]{-1+x^2}\right )-\frac {1}{2} \log \left (1+2 x+x^2+(1+x) \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \]
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\[ \int \frac {-3+x}{\sqrt [3]{-1+x^2} \left (2+x+x^2\right )} \, dx=\int \frac {-3+x}{\sqrt [3]{-1+x^2} \left (2+x+x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-3+x}{\sqrt [3]{-1+x^2} \left (2+x+x^2\right )} \, dx \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {-3+x}{\sqrt [3]{-1+x^2} \left (2+x+x^2\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{2+2 x+\sqrt [3]{-1+x^2}}\right )+\log \left (-1-x+\sqrt [3]{-1+x^2}\right )-\frac {1}{2} \log \left (1+2 x+x^2+(1+x) \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.41 (sec) , antiderivative size = 334, normalized size of antiderivative = 3.75
method | result | size |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \left (x^{2}-1\right )^{\frac {2}{3}}+3 x \left (x^{2}-1\right )^{\frac {1}{3}}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +3 \left (x^{2}-1\right )^{\frac {1}{3}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-x +1}{x^{2}+x +2}\right )-\ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \left (x^{2}-1\right )^{\frac {2}{3}}+3 x \left (x^{2}-1\right )^{\frac {1}{3}}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -x^{2}+3 \left (x^{2}-1\right )^{\frac {1}{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 x -1}{x^{2}+x +2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \left (x^{2}-1\right )^{\frac {2}{3}}+3 x \left (x^{2}-1\right )^{\frac {1}{3}}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -x^{2}+3 \left (x^{2}-1\right )^{\frac {1}{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 x -1}{x^{2}+x +2}\right )\) | \(334\) |
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Time = 0.55 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07 \[ \int \frac {-3+x}{\sqrt [3]{-1+x^2} \left (2+x+x^2\right )} \, dx=-\sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + \sqrt {3} {\left (x - 1\right )} - 2 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{8 \, x^{2} + 17 \, x + 7}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} - 3 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + x + 3 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} + 2}{x^{2} + x + 2}\right ) \]
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\[ \int \frac {-3+x}{\sqrt [3]{-1+x^2} \left (2+x+x^2\right )} \, dx=\int \frac {x - 3}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + x + 2\right )}\, dx \]
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\[ \int \frac {-3+x}{\sqrt [3]{-1+x^2} \left (2+x+x^2\right )} \, dx=\int { \frac {x - 3}{{\left (x^{2} + x + 2\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {-3+x}{\sqrt [3]{-1+x^2} \left (2+x+x^2\right )} \, dx=\int { \frac {x - 3}{{\left (x^{2} + x + 2\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {-3+x}{\sqrt [3]{-1+x^2} \left (2+x+x^2\right )} \, dx=\int \frac {x-3}{{\left (x^2-1\right )}^{1/3}\,\left (x^2+x+2\right )} \,d x \]
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