Integrand size = 132, antiderivative size = 34 \[ \int \frac {-2 x^2+\left (e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))+\left (-2 x^2+\left (2 e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))\right ) \log \left (\frac {x^2+\left (-e^3 x-x^2\right ) (i \pi +\log (4+3 e))}{i \pi +\log (4+3 e)}\right )}{-x+\left (e^3+x\right ) (i \pi +\log (4+3 e))} \, dx=x^2 \log \left (x \left (-e^3-x+\frac {x}{i \pi +\log (-5+3 (3+e))}\right )\right ) \]
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Time = 0.89 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6873, 6820, 6874, 78, 2581, 30, 45} \[ \int \frac {-2 x^2+\left (e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))+\left (-2 x^2+\left (2 e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))\right ) \log \left (\frac {x^2+\left (-e^3 x-x^2\right ) (i \pi +\log (4+3 e))}{i \pi +\log (4+3 e)}\right )}{-x+\left (e^3+x\right ) (i \pi +\log (4+3 e))} \, dx=x^2 \log \left (-x \left (e^3+x \left (1-\frac {1}{\log (4+3 e)+i \pi }\right )\right )\right ) \]
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Rule 30
Rule 45
Rule 78
Rule 2581
Rule 6820
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x^2-\left (e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))-\left (-2 x^2+\left (2 e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))\right ) \log \left (\frac {x^2+\left (-e^3 x-x^2\right ) (i \pi +\log (4+3 e))}{i \pi +\log (4+3 e)}\right )}{x (1-i \pi -\log (4+3 e))-e^3 (i \pi +\log (4+3 e))} \, dx \\ & = \int \frac {x \left (2 x-\left (e^3+2 x\right ) (i \pi +\log (4+3 e))-2 \left (-x+\left (e^3+x\right ) (i \pi +\log (4+3 e))\right ) \log \left (x \left (-e^3+x \left (-1+\frac {1}{i \pi +\log (4+3 e)}\right )\right )\right )\right )}{x (1-i \pi -\log (4+3 e))-e^3 (i \pi +\log (4+3 e))} \, dx \\ & = \int \left (\frac {x \left (2 x (\pi +i (1-\log (4+3 e)))+e^3 (\pi -i \log (4+3 e))\right )}{x (\pi +i (1-\log (4+3 e)))+e^3 (\pi -i \log (4+3 e))}+2 x \log \left (x \left (-e^3-x \left (1-\frac {1}{i \pi +\log (4+3 e)}\right )\right )\right )\right ) \, dx \\ & = 2 \int x \log \left (x \left (-e^3+x \left (-1+\frac {1}{i \pi +\log (4+3 e)}\right )\right )\right ) \, dx+\int \frac {x \left (2 x (\pi +i (1-\log (4+3 e)))+e^3 (\pi -i \log (4+3 e))\right )}{x (\pi +i (1-\log (4+3 e)))+e^3 (\pi -i \log (4+3 e))} \, dx \\ & = x^2 \log \left (-x \left (e^3+x \left (1-\frac {1}{i \pi +\log (4+3 e)}\right )\right )\right )+\left (1-\frac {1}{i \pi +\log (4+3 e)}\right ) \int \frac {x^2}{-e^3+x \left (-1+\frac {1}{i \pi +\log (4+3 e)}\right )} \, dx-\int x \, dx+\int \left (2 x+\frac {e^6 (\pi -i \log (4+3 e))^2}{(\pi +i (1-\log (4+3 e))) \left (x (\pi +i (1-\log (4+3 e)))+e^3 (\pi -i \log (4+3 e))\right )}+\frac {e^3 (-\pi +i \log (4+3 e))}{\pi +i (1-\log (4+3 e))}\right ) \, dx \\ & = \frac {x^2}{2}-\frac {e^3 x (\pi -i \log (4+3 e))}{\pi +i (1-\log (4+3 e))}+\frac {e^6 (\pi -i \log (4+3 e))^2 \log \left (x (\pi +i (1-\log (4+3 e)))+e^3 (\pi -i \log (4+3 e))\right )}{(\pi +i (1-\log (4+3 e)))^2}+x^2 \log \left (-x \left (e^3+x \left (1-\frac {1}{i \pi +\log (4+3 e)}\right )\right )\right )+\left (1-\frac {1}{i \pi +\log (4+3 e)}\right ) \int \left (\frac {e^3 (\pi -i \log (4+3 e))^2}{(\pi +i (1-\log (4+3 e)))^2}+\frac {e^6 (\pi -i \log (4+3 e))^3}{(\pi +i (1-\log (4+3 e)))^2 \left (-x (\pi +i (1-\log (4+3 e)))-e^3 (\pi -i \log (4+3 e))\right )}+\frac {x (-\pi +i \log (4+3 e))}{\pi +i (1-\log (4+3 e))}\right ) \, dx \\ & = x^2 \log \left (-x \left (e^3+x \left (1-\frac {1}{i \pi +\log (4+3 e)}\right )\right )\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {-2 x^2+\left (e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))+\left (-2 x^2+\left (2 e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))\right ) \log \left (\frac {x^2+\left (-e^3 x-x^2\right ) (i \pi +\log (4+3 e))}{i \pi +\log (4+3 e)}\right )}{-x+\left (e^3+x\right ) (i \pi +\log (4+3 e))} \, dx=x^2 \log \left (x \left (-e^3+x \left (-1+\frac {1}{i \pi +\log (4+3 e)}\right )\right )\right ) \]
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Time = 3.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15
method | result | size |
norman | \(x^{2} \ln \left (\frac {\left (-x \,{\mathrm e}^{3}-x^{2}\right ) \ln \left (-3 \,{\mathrm e}-4\right )+x^{2}}{\ln \left (-3 \,{\mathrm e}-4\right )}\right )\) | \(39\) |
risch | \(x^{2} \ln \left (\frac {\left (-x \,{\mathrm e}^{3}-x^{2}\right ) \ln \left (-3 \,{\mathrm e}-4\right )+x^{2}}{\ln \left (-3 \,{\mathrm e}-4\right )}\right )\) | \(39\) |
default | \(x^{2} \ln \left (\frac {x \left (-{\mathrm e}^{3} \ln \left (-3 \,{\mathrm e}-4\right )-\ln \left (-3 \,{\mathrm e}-4\right ) x +x \right )}{\ln \left (-3 \,{\mathrm e}-4\right )}\right )\) | \(40\) |
parts | \(x^{2} \ln \left (\frac {x \left (-{\mathrm e}^{3} \ln \left (-3 \,{\mathrm e}-4\right )-\ln \left (-3 \,{\mathrm e}-4\right ) x +x \right )}{\ln \left (-3 \,{\mathrm e}-4\right )}\right )\) | \(40\) |
parallelrisch | \(\frac {\ln \left (-3 \,{\mathrm e}-4\right )^{2} x^{2} \ln \left (\frac {\left (-x \,{\mathrm e}^{3}-x^{2}\right ) \ln \left (-3 \,{\mathrm e}-4\right )+x^{2}}{\ln \left (-3 \,{\mathrm e}-4\right )}\right )-2 \ln \left (-3 \,{\mathrm e}-4\right ) x^{2} \ln \left (\frac {\left (-x \,{\mathrm e}^{3}-x^{2}\right ) \ln \left (-3 \,{\mathrm e}-4\right )+x^{2}}{\ln \left (-3 \,{\mathrm e}-4\right )}\right )+x^{2} \ln \left (\frac {\left (-x \,{\mathrm e}^{3}-x^{2}\right ) \ln \left (-3 \,{\mathrm e}-4\right )+x^{2}}{\ln \left (-3 \,{\mathrm e}-4\right )}\right )}{\left (\ln \left (-3 \,{\mathrm e}-4\right )-1\right )^{2}}\) | \(145\) |
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none
Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {-2 x^2+\left (e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))+\left (-2 x^2+\left (2 e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))\right ) \log \left (\frac {x^2+\left (-e^3 x-x^2\right ) (i \pi +\log (4+3 e))}{i \pi +\log (4+3 e)}\right )}{-x+\left (e^3+x\right ) (i \pi +\log (4+3 e))} \, dx=x^{2} \log \left (\frac {x^{2} - {\left (x^{2} + x e^{3}\right )} \log \left (-3 \, e - 4\right )}{\log \left (-3 \, e - 4\right )}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (24) = 48\).
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.15 \[ \int \frac {-2 x^2+\left (e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))+\left (-2 x^2+\left (2 e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))\right ) \log \left (\frac {x^2+\left (-e^3 x-x^2\right ) (i \pi +\log (4+3 e))}{i \pi +\log (4+3 e)}\right )}{-x+\left (e^3+x\right ) (i \pi +\log (4+3 e))} \, dx=x^{2} \log {\left (- \frac {i \pi x^{2}}{\log {\left (4 + 3 e \right )} + i \pi } + \frac {x^{2}}{\log {\left (4 + 3 e \right )} + i \pi } - \frac {x^{2} \log {\left (4 + 3 e \right )}}{\log {\left (4 + 3 e \right )} + i \pi } - \frac {i \pi x e^{3}}{\log {\left (4 + 3 e \right )} + i \pi } - \frac {x e^{3} \log {\left (4 + 3 e \right )}}{\log {\left (4 + 3 e \right )} + i \pi } \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 689 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 689, normalized size of antiderivative = 20.26 \[ \int \frac {-2 x^2+\left (e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))+\left (-2 x^2+\left (2 e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))\right ) \log \left (\frac {x^2+\left (-e^3 x-x^2\right ) (i \pi +\log (4+3 e))}{i \pi +\log (4+3 e)}\right )}{-x+\left (e^3+x\right ) (i \pi +\log (4+3 e))} \, dx=-{\left (\frac {e^{3} \log \left (x {\left (\log \left (-3 \, e - 4\right ) - 1\right )} + e^{3} \log \left (-3 \, e - 4\right )\right ) \log \left (-3 \, e - 4\right )}{\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1} - \frac {x}{\log \left (-3 \, e - 4\right ) - 1}\right )} e^{3} \log \left (-3 \, e - 4\right ) - \frac {2 \, e^{6} \log \left (x {\left (\log \left (-3 \, e - 4\right ) - 1\right )} + e^{3} \log \left (-3 \, e - 4\right )\right ) \log \left (-3 \, e - 4\right )^{2}}{\log \left (-3 \, e - 4\right )^{3} - 3 \, \log \left (-3 \, e - 4\right )^{2} + 3 \, \log \left (-3 \, e - 4\right ) - 1} + {\left (\frac {2 \, e^{6} \log \left (x {\left (\log \left (-3 \, e - 4\right ) - 1\right )} + e^{3} \log \left (-3 \, e - 4\right )\right ) \log \left (-3 \, e - 4\right )^{2}}{\log \left (-3 \, e - 4\right )^{3} - 3 \, \log \left (-3 \, e - 4\right )^{2} + 3 \, \log \left (-3 \, e - 4\right ) - 1} + \frac {x^{2} {\left (\log \left (-3 \, e - 4\right ) - 1\right )} - 2 \, x e^{3} \log \left (-3 \, e - 4\right )}{\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1}\right )} \log \left (-3 \, e - 4\right ) + \frac {{\left ({\left (\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} \log \left (-3 \, e - 4\right ) - \log \left (-3 \, e - 4\right )^{2} + 2 \, \log \left (-3 \, e - 4\right ) - 1\right )} x^{2} \log \left (x\right ) + {\left (\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} x e^{3} \log \left (-3 \, e - 4\right ) - {\left ({\left (\log \left (-3 \, e - 4\right )^{2} + {\left (\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} \log \left (\log \left (-3 \, e - 4\right )\right ) - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} \log \left (-3 \, e - 4\right ) - \log \left (-3 \, e - 4\right )^{2} - {\left (\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} \log \left (\log \left (-3 \, e - 4\right )\right ) + 2 \, \log \left (-3 \, e - 4\right ) - 1\right )} x^{2} + {\left ({\left ({\left (\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} \log \left (-3 \, e - 4\right ) - \log \left (-3 \, e - 4\right )^{2} + 2 \, \log \left (-3 \, e - 4\right ) - 1\right )} x^{2} - {\left (\log \left (-3 \, e - 4\right )^{3} - \log \left (-3 \, e - 4\right )^{2}\right )} e^{6}\right )} \log \left (-x {\left (\log \left (-3 \, e - 4\right ) - 1\right )} - e^{3} \log \left (-3 \, e - 4\right )\right )}{{\left (\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} \log \left (-3 \, e - 4\right ) - \log \left (-3 \, e - 4\right )^{2} + 2 \, \log \left (-3 \, e - 4\right ) - 1} - \frac {x^{2} {\left (\log \left (-3 \, e - 4\right ) - 1\right )} - 2 \, x e^{3} \log \left (-3 \, e - 4\right )}{\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1346 vs. \(2 (24) = 48\).
Time = 0.49 (sec) , antiderivative size = 1346, normalized size of antiderivative = 39.59 \[ \int \frac {-2 x^2+\left (e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))+\left (-2 x^2+\left (2 e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))\right ) \log \left (\frac {x^2+\left (-e^3 x-x^2\right ) (i \pi +\log (4+3 e))}{i \pi +\log (4+3 e)}\right )}{-x+\left (e^3+x\right ) (i \pi +\log (4+3 e))} \, dx=\text {Too large to display} \]
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Time = 13.79 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int \frac {-2 x^2+\left (e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))+\left (-2 x^2+\left (2 e^3 x+2 x^2\right ) (i \pi +\log (4+3 e))\right ) \log \left (\frac {x^2+\left (-e^3 x-x^2\right ) (i \pi +\log (4+3 e))}{i \pi +\log (4+3 e)}\right )}{-x+\left (e^3+x\right ) (i \pi +\log (4+3 e))} \, dx=x^2\,\ln \left (-\frac {\ln \left (-3\,\mathrm {e}-4\right )\,\left (x^2+{\mathrm {e}}^3\,x\right )-x^2}{\ln \left (-3\,\mathrm {e}-4\right )}\right ) \]
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