Integrand size = 144, antiderivative size = 24 \[ \int \frac {e^{\frac {2+8 x+2 x^2+\left (x+2 x^2+x^3\right ) \log \left (\frac {1}{x}\right )}{2+2 x+\left (x+x^2\right ) \log \left (\frac {1}{x}\right )}} \left (12+12 x+8 x^2+\left (4 x+4 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )\right )}{4+8 x+4 x^2+\left (4 x+8 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=e^{1+x+\frac {4}{(1+x) \left (\frac {2}{x}+\log \left (\frac {1}{x}\right )\right )}} \]
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\[ \int \frac {e^{\frac {2+8 x+2 x^2+\left (x+2 x^2+x^3\right ) \log \left (\frac {1}{x}\right )}{2+2 x+\left (x+x^2\right ) \log \left (\frac {1}{x}\right )}} \left (12+12 x+8 x^2+\left (4 x+4 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )\right )}{4+8 x+4 x^2+\left (4 x+8 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=\int \frac {\exp \left (\frac {2+8 x+2 x^2+\left (x+2 x^2+x^3\right ) \log \left (\frac {1}{x}\right )}{2+2 x+\left (x+x^2\right ) \log \left (\frac {1}{x}\right )}\right ) \left (12+12 x+8 x^2+\left (4 x+4 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )\right )}{4+8 x+4 x^2+\left (4 x+8 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}} \left (4 \left (3+3 x+2 x^2\right )+4 x \left (1+x+x^2\right ) \log \left (\frac {1}{x}\right )+x^2 (1+x)^2 \log ^2\left (\frac {1}{x}\right )\right )}{(1+x)^2 \left (2+x \log \left (\frac {1}{x}\right )\right )^2} \, dx \\ & = \int \left (\exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}}+\frac {4 \exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}} (2+x)}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )^2}-\frac {4 \exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{-1+\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}}}{(1+x)^2 \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \, dx \\ & = 4 \int \frac {\exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}} (2+x)}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )^2} \, dx-4 \int \frac {\exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{-1+\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}}}{(1+x)^2 \left (2+x \log \left (\frac {1}{x}\right )\right )} \, dx+\int \exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}} \, dx \\ & = -\left (4 \int \frac {\exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{-1+\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}}}{(1+x)^2 \left (2+x \log \left (\frac {1}{x}\right )\right )} \, dx\right )+4 \int \left (\frac {\exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}}}{\left (2+x \log \left (\frac {1}{x}\right )\right )^2}+\frac {\exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}}}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )^2}\right ) \, dx+\int \exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}} \, dx \\ & = 4 \int \frac {\exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}}}{\left (2+x \log \left (\frac {1}{x}\right )\right )^2} \, dx+4 \int \frac {\exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}}}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )^2} \, dx-4 \int \frac {\exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{-1+\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}}}{(1+x)^2 \left (2+x \log \left (\frac {1}{x}\right )\right )} \, dx+\int \exp \left (\frac {2 \left (1+4 x+x^2\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}\right ) \left (\frac {1}{x}\right )^{\frac {x (1+x)}{2+x \log \left (\frac {1}{x}\right )}} \, dx \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {e^{\frac {2+8 x+2 x^2+\left (x+2 x^2+x^3\right ) \log \left (\frac {1}{x}\right )}{2+2 x+\left (x+x^2\right ) \log \left (\frac {1}{x}\right )}} \left (12+12 x+8 x^2+\left (4 x+4 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )\right )}{4+8 x+4 x^2+\left (4 x+8 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=e^{\frac {2 \left (1+4 x+x^2\right )+x (1+x)^2 \log \left (\frac {1}{x}\right )}{(1+x) \left (2+x \log \left (\frac {1}{x}\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(23)=46\).
Time = 8.81 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {\left (x^{3}+2 x^{2}+x \right ) \ln \left (\frac {1}{x}\right )+2 x^{2}+8 x +2}{x^{2} \ln \left (\frac {1}{x}\right )+x \ln \left (\frac {1}{x}\right )+2 x +2}}\) | \(49\) |
risch | \({\mathrm e}^{\frac {x^{3} \ln \left (\frac {1}{x}\right )+2 x^{2} \ln \left (\frac {1}{x}\right )+x \ln \left (\frac {1}{x}\right )+2 x^{2}+8 x +2}{\left (1+x \right ) \left (x \ln \left (\frac {1}{x}\right )+2\right )}}\) | \(51\) |
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Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {e^{\frac {2+8 x+2 x^2+\left (x+2 x^2+x^3\right ) \log \left (\frac {1}{x}\right )}{2+2 x+\left (x+x^2\right ) \log \left (\frac {1}{x}\right )}} \left (12+12 x+8 x^2+\left (4 x+4 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )\right )}{4+8 x+4 x^2+\left (4 x+8 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=e^{\left (\frac {2 \, x^{2} + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \left (\frac {1}{x}\right ) + 8 \, x + 2}{{\left (x^{2} + x\right )} \log \left (\frac {1}{x}\right ) + 2 \, x + 2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (17) = 34\).
Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {e^{\frac {2+8 x+2 x^2+\left (x+2 x^2+x^3\right ) \log \left (\frac {1}{x}\right )}{2+2 x+\left (x+x^2\right ) \log \left (\frac {1}{x}\right )}} \left (12+12 x+8 x^2+\left (4 x+4 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )\right )}{4+8 x+4 x^2+\left (4 x+8 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=e^{\frac {2 x^{2} + 8 x + \left (x^{3} + 2 x^{2} + x\right ) \log {\left (\frac {1}{x} \right )} + 2}{2 x + \left (x^{2} + x\right ) \log {\left (\frac {1}{x} \right )} + 2}} \]
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Time = 0.41 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {e^{\frac {2+8 x+2 x^2+\left (x+2 x^2+x^3\right ) \log \left (\frac {1}{x}\right )}{2+2 x+\left (x+x^2\right ) \log \left (\frac {1}{x}\right )}} \left (12+12 x+8 x^2+\left (4 x+4 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )\right )}{4+8 x+4 x^2+\left (4 x+8 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=e^{\left (x - \frac {8}{x \log \left (x\right )^{2} + 2 \, {\left (x - 1\right )} \log \left (x\right ) - 4} - \frac {4}{{\left (x + 1\right )} \log \left (x\right ) + 2 \, x + 2} + 1\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (23) = 46\).
Time = 0.46 (sec) , antiderivative size = 131, normalized size of antiderivative = 5.46 \[ \int \frac {e^{\frac {2+8 x+2 x^2+\left (x+2 x^2+x^3\right ) \log \left (\frac {1}{x}\right )}{2+2 x+\left (x+x^2\right ) \log \left (\frac {1}{x}\right )}} \left (12+12 x+8 x^2+\left (4 x+4 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )\right )}{4+8 x+4 x^2+\left (4 x+8 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=e^{\left (\frac {x^{3} \log \left (x\right )}{x^{2} \log \left (x\right ) + x \log \left (x\right ) - 2 \, x - 2} + \frac {2 \, x^{2} \log \left (x\right )}{x^{2} \log \left (x\right ) + x \log \left (x\right ) - 2 \, x - 2} - \frac {2 \, x^{2}}{x^{2} \log \left (x\right ) + x \log \left (x\right ) - 2 \, x - 2} + \frac {x \log \left (x\right )}{x^{2} \log \left (x\right ) + x \log \left (x\right ) - 2 \, x - 2} - \frac {8 \, x}{x^{2} \log \left (x\right ) + x \log \left (x\right ) - 2 \, x - 2} - \frac {2}{x^{2} \log \left (x\right ) + x \log \left (x\right ) - 2 \, x - 2}\right )} \]
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Time = 12.90 (sec) , antiderivative size = 97, normalized size of antiderivative = 4.04 \[ \int \frac {e^{\frac {2+8 x+2 x^2+\left (x+2 x^2+x^3\right ) \log \left (\frac {1}{x}\right )}{2+2 x+\left (x+x^2\right ) \log \left (\frac {1}{x}\right )}} \left (12+12 x+8 x^2+\left (4 x+4 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )\right )}{4+8 x+4 x^2+\left (4 x+8 x^2+4 x^3\right ) \log \left (\frac {1}{x}\right )+\left (x^2+2 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx={\mathrm {e}}^{\frac {8\,x}{2\,x+x\,\ln \left (\frac {1}{x}\right )+x^2\,\ln \left (\frac {1}{x}\right )+2}}\,{\mathrm {e}}^{\frac {2\,x^2}{2\,x+x\,\ln \left (\frac {1}{x}\right )+x^2\,\ln \left (\frac {1}{x}\right )+2}}\,{\mathrm {e}}^{\frac {2}{2\,x+x\,\ln \left (\frac {1}{x}\right )+x^2\,\ln \left (\frac {1}{x}\right )+2}}\,{\left (\frac {1}{x}\right )}^{\frac {x^2+x}{x\,\ln \left (\frac {1}{x}\right )+2}} \]
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