Integrand size = 145, antiderivative size = 30 \[ \int \frac {-2 x^2+e^6 \left (-x+x^2\right )+e^3 \left (-2 x^2+x^3\right )+\left (x+e^3 x+x^2-3 x^3+x^4\right ) \log (x)+2 x^2 \log ^2(x)+\left (e^6+x+e^3 x+\left (-e^6-x-e^3 x+x^2\right ) \log (x)-x^2 \log ^2(x)\right ) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{e^6 x^2+x^3+e^3 x^3+x^4 \log (x)} \, dx=\frac {(-x+\log (x)) \left (-x+\log \left (e^6+x+x \left (e^3+x \log (x)\right )\right )\right )}{x} \]
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\[ \int \frac {-2 x^2+e^6 \left (-x+x^2\right )+e^3 \left (-2 x^2+x^3\right )+\left (x+e^3 x+x^2-3 x^3+x^4\right ) \log (x)+2 x^2 \log ^2(x)+\left (e^6+x+e^3 x+\left (-e^6-x-e^3 x+x^2\right ) \log (x)-x^2 \log ^2(x)\right ) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{e^6 x^2+x^3+e^3 x^3+x^4 \log (x)} \, dx=\int \frac {-2 x^2+e^6 \left (-x+x^2\right )+e^3 \left (-2 x^2+x^3\right )+\left (x+e^3 x+x^2-3 x^3+x^4\right ) \log (x)+2 x^2 \log ^2(x)+\left (e^6+x+e^3 x+\left (-e^6-x-e^3 x+x^2\right ) \log (x)-x^2 \log ^2(x)\right ) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{e^6 x^2+x^3+e^3 x^3+x^4 \log (x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-2 x^2+e^6 \left (-x+x^2\right )+e^3 \left (-2 x^2+x^3\right )+\left (x+e^3 x+x^2-3 x^3+x^4\right ) \log (x)+2 x^2 \log ^2(x)+\left (e^6+x+e^3 x+\left (-e^6-x-e^3 x+x^2\right ) \log (x)-x^2 \log ^2(x)\right ) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{e^6 x^2+\left (1+e^3\right ) x^3+x^4 \log (x)} \, dx \\ & = \int \left (\frac {2}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)}+\frac {e^3 (-2+x)}{e^6+\left (1+e^3\right ) x+x^2 \log (x)}+\frac {e^6 (-1+x)}{x \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {\left (1+e^3+x-3 x^2+x^3\right ) \log (x)}{x \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {2 \log ^2(x)}{e^6+\left (1+e^3\right ) x+x^2 \log (x)}+\frac {(1-\log (x)) \log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2}\right ) \, dx \\ & = 2 \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+2 \int \frac {\log ^2(x)}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+e^3 \int \frac {-2+x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+e^6 \int \frac {-1+x}{x \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\int \frac {\left (1+e^3+x-3 x^2+x^3\right ) \log (x)}{x \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\int \frac {(1-\log (x)) \log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx \\ & = 2 \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+2 \int \left (\frac {-e^6-\left (1+e^3\right ) x}{x^4}+\frac {\log (x)}{x^2}+\frac {\left (e^6+\left (1+e^3\right ) x\right )^2}{x^4 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}\right ) \, dx+e^3 \int \left (\frac {2}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)}+\frac {x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)}\right ) \, dx+e^6 \int \left (\frac {1}{x \left (-e^6-\left (1+e^3\right ) x-x^2 \log (x)\right )}+\frac {1}{e^6+\left (1+e^3\right ) x+x^2 \log (x)}\right ) \, dx+\int \left (\frac {1+e^3+x-3 x^2+x^3}{x^3}+\frac {-e^6 \left (1+e^3\right )-\left (1+2 e^3+2 e^6\right ) x-\left (1+e^3-3 e^6\right ) x^2+\left (3+3 e^3-e^6\right ) x^3-\left (1+e^3\right ) x^4}{x^3 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}\right ) \, dx+\int \left (\frac {\log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2}-\frac {\log (x) \log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2}\right ) \, dx \\ & = 2 \int \frac {-e^6-\left (1+e^3\right ) x}{x^4} \, dx+2 \int \frac {\log (x)}{x^2} \, dx+2 \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+2 \int \frac {\left (e^6+\left (1+e^3\right ) x\right )^2}{x^4 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+e^3 \int \frac {x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+\left (2 e^3\right ) \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+e^6 \int \frac {1}{x \left (-e^6-\left (1+e^3\right ) x-x^2 \log (x)\right )} \, dx+e^6 \int \frac {1}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+\int \frac {1+e^3+x-3 x^2+x^3}{x^3} \, dx+\int \frac {-e^6 \left (1+e^3\right )-\left (1+2 e^3+2 e^6\right ) x-\left (1+e^3-3 e^6\right ) x^2+\left (3+3 e^3-e^6\right ) x^3-\left (1+e^3\right ) x^4}{x^3 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\int \frac {\log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx-\int \frac {\log (x) \log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx \\ & = -\frac {2}{x}-\frac {2 \log (x)}{x}+2 \int \left (-\frac {e^6}{x^4}+\frac {-1-e^3}{x^3}\right ) \, dx+2 \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+2 \int \left (\frac {e^{12}}{x^4 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {2 \left (e^6+e^9\right )}{x^3 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {\left (1+e^3\right )^2}{x^2 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}\right ) \, dx+e^3 \int \frac {x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+\left (2 e^3\right ) \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+e^6 \int \frac {1}{x \left (-e^6-\left (1+e^3\right ) x-x^2 \log (x)\right )} \, dx+e^6 \int \frac {1}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+\int \left (1+\frac {1+e^3}{x^3}+\frac {1}{x^2}-\frac {3}{x}\right ) \, dx+\int \left (\frac {e^6 \left (1-\frac {3 \left (1+e^3\right )}{e^6}\right )}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)}+\frac {e^6 \left (-1-e^3\right )}{x^3 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {-1-2 e^3-2 e^6}{x^2 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {-1-e^3+3 e^6}{x \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}+\frac {\left (-1-e^3\right ) x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)}\right ) \, dx+\int \frac {\log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx-\int \frac {\log (x) \log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx \\ & = \frac {2 e^6}{3 x^3}+\frac {1+e^3}{2 x^2}-\frac {3}{x}+x-3 \log (x)-\frac {2 \log (x)}{x}+2 \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+e^3 \int \frac {x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+\left (2 e^3\right ) \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+e^6 \int \frac {1}{x \left (-e^6-\left (1+e^3\right ) x-x^2 \log (x)\right )} \, dx+e^6 \int \frac {1}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx+\left (2 e^{12}\right ) \int \frac {1}{x^4 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\left (-1-e^3\right ) \int \frac {x}{e^6+\left (1+e^3\right ) x+x^2 \log (x)} \, dx-\left (e^6 \left (1+e^3\right )\right ) \int \frac {1}{x^3 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\left (4 e^6 \left (1+e^3\right )\right ) \int \frac {1}{x^3 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\left (2 \left (1+e^3\right )^2\right ) \int \frac {1}{x^2 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\left (-1-2 e^3-2 e^6\right ) \int \frac {1}{x^2 \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\left (-3-3 e^3+e^6\right ) \int \frac {1}{-e^6-\left (1+e^3\right ) x-x^2 \log (x)} \, dx+\left (-1-e^3+3 e^6\right ) \int \frac {1}{x \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )} \, dx+\int \frac {\log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx-\int \frac {\log (x) \log \left (e^6+\left (1+e^3\right ) x+x^2 \log (x)\right )}{x^2} \, dx \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {-2 x^2+e^6 \left (-x+x^2\right )+e^3 \left (-2 x^2+x^3\right )+\left (x+e^3 x+x^2-3 x^3+x^4\right ) \log (x)+2 x^2 \log ^2(x)+\left (e^6+x+e^3 x+\left (-e^6-x-e^3 x+x^2\right ) \log (x)-x^2 \log ^2(x)\right ) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{e^6 x^2+x^3+e^3 x^3+x^4 \log (x)} \, dx=x-\log (x)-\log \left (e^6+x+e^3 x+x^2 \log (x)\right )+\frac {\log (x) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{x} \]
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Time = 2.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53
method | result | size |
risch | \(\frac {\ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )+{\mathrm e}^{6}+x \,{\mathrm e}^{3}+x \right )}{x}+x -3 \ln \left (x \right )-\ln \left (\ln \left (x \right )+\frac {x \,{\mathrm e}^{3}+{\mathrm e}^{6}+x}{x^{2}}\right )\) | \(46\) |
parallelrisch | \(\frac {x^{2}-x \ln \left (x \right )-\ln \left (x^{2} \ln \left (x \right )+{\mathrm e}^{6}+x \,{\mathrm e}^{3}+x \right ) x +\ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )+{\mathrm e}^{6}+x \,{\mathrm e}^{3}+x \right )}{x}\) | \(54\) |
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Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {-2 x^2+e^6 \left (-x+x^2\right )+e^3 \left (-2 x^2+x^3\right )+\left (x+e^3 x+x^2-3 x^3+x^4\right ) \log (x)+2 x^2 \log ^2(x)+\left (e^6+x+e^3 x+\left (-e^6-x-e^3 x+x^2\right ) \log (x)-x^2 \log ^2(x)\right ) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{e^6 x^2+x^3+e^3 x^3+x^4 \log (x)} \, dx=\frac {x^{2} - {\left (x - \log \left (x\right )\right )} \log \left (x^{2} \log \left (x\right ) + x e^{3} + x + e^{6}\right ) - x \log \left (x\right )}{x} \]
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Time = 0.31 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {-2 x^2+e^6 \left (-x+x^2\right )+e^3 \left (-2 x^2+x^3\right )+\left (x+e^3 x+x^2-3 x^3+x^4\right ) \log (x)+2 x^2 \log ^2(x)+\left (e^6+x+e^3 x+\left (-e^6-x-e^3 x+x^2\right ) \log (x)-x^2 \log ^2(x)\right ) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{e^6 x^2+x^3+e^3 x^3+x^4 \log (x)} \, dx=x - 3 \log {\left (x \right )} - \log {\left (\log {\left (x \right )} + \frac {x + x e^{3} + e^{6}}{x^{2}} \right )} + \frac {\log {\left (x \right )} \log {\left (x^{2} \log {\left (x \right )} + x + x e^{3} + e^{6} \right )}}{x} \]
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Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {-2 x^2+e^6 \left (-x+x^2\right )+e^3 \left (-2 x^2+x^3\right )+\left (x+e^3 x+x^2-3 x^3+x^4\right ) \log (x)+2 x^2 \log ^2(x)+\left (e^6+x+e^3 x+\left (-e^6-x-e^3 x+x^2\right ) \log (x)-x^2 \log ^2(x)\right ) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{e^6 x^2+x^3+e^3 x^3+x^4 \log (x)} \, dx=\frac {x^{2} + \log \left (x^{2} \log \left (x\right ) + x {\left (e^{3} + 1\right )} + e^{6}\right ) \log \left (x\right )}{x} - 3 \, \log \left (x\right ) - \log \left (\frac {x^{2} \log \left (x\right ) + x {\left (e^{3} + 1\right )} + e^{6}}{x^{2}}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.83 \[ \int \frac {-2 x^2+e^6 \left (-x+x^2\right )+e^3 \left (-2 x^2+x^3\right )+\left (x+e^3 x+x^2-3 x^3+x^4\right ) \log (x)+2 x^2 \log ^2(x)+\left (e^6+x+e^3 x+\left (-e^6-x-e^3 x+x^2\right ) \log (x)-x^2 \log ^2(x)\right ) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{e^6 x^2+x^3+e^3 x^3+x^4 \log (x)} \, dx=\frac {x^{2} - x \log \left (-x^{2} \log \left (x\right ) - x e^{3} - x - e^{6}\right ) - x \log \left (x\right ) + \log \left (x^{2} \log \left (x\right ) + x e^{3} + x + e^{6}\right ) \log \left (x\right )}{x} \]
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Time = 10.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {-2 x^2+e^6 \left (-x+x^2\right )+e^3 \left (-2 x^2+x^3\right )+\left (x+e^3 x+x^2-3 x^3+x^4\right ) \log (x)+2 x^2 \log ^2(x)+\left (e^6+x+e^3 x+\left (-e^6-x-e^3 x+x^2\right ) \log (x)-x^2 \log ^2(x)\right ) \log \left (e^6+x+e^3 x+x^2 \log (x)\right )}{e^6 x^2+x^3+e^3 x^3+x^4 \log (x)} \, dx=x-\ln \left (x+{\mathrm {e}}^6+x^2\,\ln \left (x\right )+x\,{\mathrm {e}}^3\right )-\ln \left (\frac {1}{x^2}\right )-3\,\ln \left (x\right )+\frac {\ln \left (x+{\mathrm {e}}^6+x^2\,\ln \left (x\right )+x\,{\mathrm {e}}^3\right )\,\ln \left (x\right )}{x} \]
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