Integrand size = 33, antiderivative size = 100 \[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\frac {2}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {\sqrt [3]{2+x^2}}{\sqrt {3}}}{\sqrt [3]{2+x^2}}\right )+\log \left (-1+x+\sqrt [3]{2+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{2+x^2}+\left (2+x^2\right )^{2/3}\right ) \]
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\[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx=\int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {6}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )}+\frac {2 x}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )}+\frac {x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )}\right ) \, dx \\ & = 2 \int \frac {x}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx+6 \int \frac {1}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx+\int \frac {x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.87 \[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {2-2 x+\sqrt [3]{2+x^2}}{\sqrt {3} \sqrt [3]{2+x^2}}\right )+\log \left (-1+x+\sqrt [3]{2+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2-(-1+x) \sqrt [3]{2+x^2}+\left (2+x^2\right )^{2/3}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.92 (sec) , antiderivative size = 511, normalized size of antiderivative = 5.11
method | result | size |
trager | \(\ln \left (\frac {394 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-985 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-573 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}} x +318 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}+797 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+1182 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +573 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}}-636 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x -2682 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-891 \left (x^{2}+2\right )^{\frac {2}{3}} x -573 \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}+309 x^{3}+318 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}}+2391 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +891 \left (x^{2}+2\right )^{\frac {2}{3}}+1146 \left (x^{2}+2\right )^{\frac {1}{3}} x -1133 x^{2}-1379 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-573 \left (x^{2}+2\right )^{\frac {1}{3}}+927 x -721}{x^{3}-2 x^{2}+3 x +1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {206 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-515 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-573 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}} x -891 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}+291 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+618 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +573 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}}+1782 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x -367 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+318 \left (x^{2}+2\right )^{\frac {2}{3}} x -573 \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}-197 x^{3}-891 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}}+873 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -318 \left (x^{2}+2\right )^{\frac {2}{3}}+1146 \left (x^{2}+2\right )^{\frac {1}{3}} x +1182 x^{2}+721 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-573 \left (x^{2}+2\right )^{\frac {1}{3}}-591 x +1379}{x^{3}-2 x^{2}+3 x +1}\right )\) | \(511\) |
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Time = 0.71 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.38 \[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{2} + 2\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{2} - 2 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{3} - 2 \, x^{2} + 3 \, x + 1\right )}}{3 \, {\left (x^{3} - 4 \, x^{2} + 3 \, x - 3\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} - 2 \, x^{2} + 3 \, {\left (x^{2} + 2\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 3 \, {\left (x^{2} - 2 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}} + 3 \, x + 1}{x^{3} - 2 \, x^{2} + 3 \, x + 1}\right ) \]
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Timed out. \[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx=\int { \frac {x^{2} + 2 \, x + 6}{{\left (x^{3} - 2 \, x^{2} + 3 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx=\int { \frac {x^{2} + 2 \, x + 6}{{\left (x^{3} - 2 \, x^{2} + 3 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx=\int \frac {x^2+2\,x+6}{{\left (x^2+2\right )}^{1/3}\,\left (x^3-2\,x^2+3\,x+1\right )} \,d x \]
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