Integrand size = 122, antiderivative size = 31 \[ \int \frac {e^{\frac {x^4-2 x^5+3 x^6-2 x^7+x^8}{x^2-2 x \log (x)+\log ^2(x)}} \left (2 x^3-2 x^4+8 x^6-8 x^7+6 x^8+\left (-4 x^3+10 x^4-18 x^5+14 x^6-8 x^7\right ) \log (x)\right )}{-x^3+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)} \, dx=1-e^{\frac {x^4 \left (-x+x \left (\frac {1}{x}+x\right )\right )^2}{(x-\log (x))^2}} \]
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\[ \int \frac {e^{\frac {x^4-2 x^5+3 x^6-2 x^7+x^8}{x^2-2 x \log (x)+\log ^2(x)}} \left (2 x^3-2 x^4+8 x^6-8 x^7+6 x^8+\left (-4 x^3+10 x^4-18 x^5+14 x^6-8 x^7\right ) \log (x)\right )}{-x^3+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)} \, dx=\int \frac {\exp \left (\frac {x^4-2 x^5+3 x^6-2 x^7+x^8}{x^2-2 x \log (x)+\log ^2(x)}\right ) \left (2 x^3-2 x^4+8 x^6-8 x^7+6 x^8+\left (-4 x^3+10 x^4-18 x^5+14 x^6-8 x^7\right ) \log (x)\right )}{-x^3+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3 \left (1-x+x^2\right ) \left (-1+x^2-3 x^3-\left (-2+3 x-4 x^2\right ) \log (x)\right )}{(x-\log (x))^3} \, dx \\ & = 2 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3 \left (1-x+x^2\right ) \left (-1+x^2-3 x^3-\left (-2+3 x-4 x^2\right ) \log (x)\right )}{(x-\log (x))^3} \, dx \\ & = 2 \int \left (\frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} (-1+x) x^3 \left (1-x+x^2\right )^2}{(x-\log (x))^3}-\frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3 \left (2-5 x+9 x^2-7 x^3+4 x^4\right )}{(x-\log (x))^2}\right ) \, dx \\ & = 2 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} (-1+x) x^3 \left (1-x+x^2\right )^2}{(x-\log (x))^3} \, dx-2 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3 \left (2-5 x+9 x^2-7 x^3+4 x^4\right )}{(x-\log (x))^2} \, dx \\ & = 2 \int \left (-\frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3}{(x-\log (x))^3}+\frac {3 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^4}{(x-\log (x))^3}-\frac {5 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^5}{(x-\log (x))^3}+\frac {5 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^6}{(x-\log (x))^3}-\frac {3 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^7}{(x-\log (x))^3}+\frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^8}{(x-\log (x))^3}\right ) \, dx-2 \int \left (\frac {2 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3}{(x-\log (x))^2}-\frac {5 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^4}{(x-\log (x))^2}+\frac {9 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^5}{(x-\log (x))^2}-\frac {7 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^6}{(x-\log (x))^2}+\frac {4 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^7}{(x-\log (x))^2}\right ) \, dx \\ & = -\left (2 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3}{(x-\log (x))^3} \, dx\right )+2 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^8}{(x-\log (x))^3} \, dx-4 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3}{(x-\log (x))^2} \, dx+6 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^4}{(x-\log (x))^3} \, dx-6 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^7}{(x-\log (x))^3} \, dx-8 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^7}{(x-\log (x))^2} \, dx-10 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^5}{(x-\log (x))^3} \, dx+10 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^6}{(x-\log (x))^3} \, dx+10 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^4}{(x-\log (x))^2} \, dx+14 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^6}{(x-\log (x))^2} \, dx-18 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^5}{(x-\log (x))^2} \, dx \\ \end{align*}
Time = 1.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {e^{\frac {x^4-2 x^5+3 x^6-2 x^7+x^8}{x^2-2 x \log (x)+\log ^2(x)}} \left (2 x^3-2 x^4+8 x^6-8 x^7+6 x^8+\left (-4 x^3+10 x^4-18 x^5+14 x^6-8 x^7\right ) \log (x)\right )}{-x^3+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)} \, dx=-e^{\frac {x^4 \left (1-x+x^2\right )^2}{(-x+\log (x))^2}} \]
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Time = 2.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84
method | result | size |
risch | \(-{\mathrm e}^{\frac {x^{4} \left (x^{2}-x +1\right )^{2}}{\left (\ln \left (x \right )-x \right )^{2}}}\) | \(26\) |
parallelrisch | \(-{\mathrm e}^{\frac {x^{4} \left (x^{4}-2 x^{3}+3 x^{2}-2 x +1\right )}{\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}}}\) | \(41\) |
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Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {x^4-2 x^5+3 x^6-2 x^7+x^8}{x^2-2 x \log (x)+\log ^2(x)}} \left (2 x^3-2 x^4+8 x^6-8 x^7+6 x^8+\left (-4 x^3+10 x^4-18 x^5+14 x^6-8 x^7\right ) \log (x)\right )}{-x^3+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)} \, dx=-e^{\left (\frac {x^{8} - 2 \, x^{7} + 3 \, x^{6} - 2 \, x^{5} + x^{4}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}}\right )} \]
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Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\frac {x^4-2 x^5+3 x^6-2 x^7+x^8}{x^2-2 x \log (x)+\log ^2(x)}} \left (2 x^3-2 x^4+8 x^6-8 x^7+6 x^8+\left (-4 x^3+10 x^4-18 x^5+14 x^6-8 x^7\right ) \log (x)\right )}{-x^3+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)} \, dx=- e^{\frac {x^{8} - 2 x^{7} + 3 x^{6} - 2 x^{5} + x^{4}}{x^{2} - 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (30) = 60\).
Time = 1.25 (sec) , antiderivative size = 343, normalized size of antiderivative = 11.06 \[ \int \frac {e^{\frac {x^4-2 x^5+3 x^6-2 x^7+x^8}{x^2-2 x \log (x)+\log ^2(x)}} \left (2 x^3-2 x^4+8 x^6-8 x^7+6 x^8+\left (-4 x^3+10 x^4-18 x^5+14 x^6-8 x^7\right ) \log (x)\right )}{-x^3+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)} \, dx=-e^{\left (\frac {\log \left (x\right )^{8}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} + x^{6} + 2 \, x^{5} \log \left (x\right ) + 3 \, x^{4} \log \left (x\right )^{2} + 4 \, x^{3} \log \left (x\right )^{3} + 5 \, x^{2} \log \left (x\right )^{4} + 6 \, x \log \left (x\right )^{5} + 7 \, \log \left (x\right )^{6} - \frac {2 \, \log \left (x\right )^{7}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} + \frac {8 \, \log \left (x\right )^{7}}{x - \log \left (x\right )} - 2 \, x^{5} - 4 \, x^{4} \log \left (x\right ) - 6 \, x^{3} \log \left (x\right )^{2} - 8 \, x^{2} \log \left (x\right )^{3} - 10 \, x \log \left (x\right )^{4} - 12 \, \log \left (x\right )^{5} + \frac {3 \, \log \left (x\right )^{6}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} - \frac {14 \, \log \left (x\right )^{6}}{x - \log \left (x\right )} + 3 \, x^{4} + 6 \, x^{3} \log \left (x\right ) + 9 \, x^{2} \log \left (x\right )^{2} + 12 \, x \log \left (x\right )^{3} + 15 \, \log \left (x\right )^{4} - \frac {2 \, \log \left (x\right )^{5}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} + \frac {18 \, \log \left (x\right )^{5}}{x - \log \left (x\right )} - 2 \, x^{3} - 4 \, x^{2} \log \left (x\right ) - 6 \, x \log \left (x\right )^{2} - 8 \, \log \left (x\right )^{3} + \frac {\log \left (x\right )^{4}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} - \frac {10 \, \log \left (x\right )^{4}}{x - \log \left (x\right )} + x^{2} + 2 \, x \log \left (x\right ) + 3 \, \log \left (x\right )^{2} + \frac {4 \, \log \left (x\right )^{3}}{x - \log \left (x\right )}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (30) = 60\).
Time = 0.37 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.29 \[ \int \frac {e^{\frac {x^4-2 x^5+3 x^6-2 x^7+x^8}{x^2-2 x \log (x)+\log ^2(x)}} \left (2 x^3-2 x^4+8 x^6-8 x^7+6 x^8+\left (-4 x^3+10 x^4-18 x^5+14 x^6-8 x^7\right ) \log (x)\right )}{-x^3+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)} \, dx=-e^{\left (\frac {x^{8}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} - \frac {2 \, x^{7}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} + \frac {3 \, x^{6}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} - \frac {2 \, x^{5}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} + \frac {x^{4}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}}\right )} \]
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Time = 9.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.39 \[ \int \frac {e^{\frac {x^4-2 x^5+3 x^6-2 x^7+x^8}{x^2-2 x \log (x)+\log ^2(x)}} \left (2 x^3-2 x^4+8 x^6-8 x^7+6 x^8+\left (-4 x^3+10 x^4-18 x^5+14 x^6-8 x^7\right ) \log (x)\right )}{-x^3+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)} \, dx=-{\mathrm {e}}^{\frac {x^4}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}}\,{\mathrm {e}}^{-\frac {2\,x^5}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}}\,{\mathrm {e}}^{\frac {x^8}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}}\,{\mathrm {e}}^{\frac {3\,x^6}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}}\,{\mathrm {e}}^{-\frac {2\,x^7}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}} \]
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