Integrand size = 110, antiderivative size = 36 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {1}{5} e^{-x+\frac {\left (x \left (-1-2 x+\frac {x^2}{2}\right )+2 \log (x)\right )^2}{x}} x \]
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\[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {1}{20} \int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{x} \, dx \\ & = \frac {1}{20} \int \left (-12 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right )-32 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x+40 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^2+36 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^3-32 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^4+5 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^5+\frac {16 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \left (2-2 x^2+x^3\right ) \log (x)}{x}-\frac {16 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \log ^2(x)}{x}\right ) \, dx \\ & = \frac {1}{4} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^5 \, dx-\frac {3}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \, dx+\frac {4}{5} \int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \left (2-2 x^2+x^3\right ) \log (x)}{x} \, dx-\frac {4}{5} \int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \log ^2(x)}{x} \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^4 \, dx+\frac {9}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^3 \, dx+2 \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^2 \, dx \\ & = \frac {1}{4} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^5 \, dx-\frac {3}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \, dx-\frac {4}{5} \int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \log ^2(x)}{x} \, dx+\frac {4}{5} \int \left (\frac {2 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \log (x)}{x}-2 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x \log (x)+\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^2 \log (x)\right ) \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^4 \, dx+\frac {9}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^3 \, dx+2 \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^2 \, dx \\ & = \frac {1}{4} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^5 \, dx-\frac {3}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \, dx+\frac {4}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^2 \log (x) \, dx-\frac {4}{5} \int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \log ^2(x)}{x} \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^4 \, dx+\frac {8}{5} \int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \log (x)}{x} \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x \log (x) \, dx+\frac {9}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^3 \, dx+2 \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^2 \, dx \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {1}{5} e^{4 x^2+3 x^3-2 x^4+\frac {x^5}{4}+\frac {4 \log ^2(x)}{x}} x^{-3+2 (-4+x) x} \]
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Time = 0.51 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {\left (x^{3}-4 x^{2}+4 \ln \left (x \right )\right ) \left (x^{3}-4 x^{2}+4 \ln \left (x \right )-4 x \right )}{4 x}} x}{5}\) | \(39\) |
parallelrisch | \(\frac {{\mathrm e}^{-\frac {-16 \ln \left (x \right )^{2}+\left (-8 x^{3}+32 x^{2}+16 x \right ) \ln \left (x \right )-x^{6}+8 x^{5}-12 x^{4}-16 x^{3}}{4 x}} x}{5}\) | \(56\) |
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Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {1}{5} \, x e^{\left (\frac {x^{6} - 8 \, x^{5} + 12 \, x^{4} + 16 \, x^{3} + 8 \, {\left (x^{3} - 4 \, x^{2} - 2 \, x\right )} \log \left (x\right ) + 16 \, \log \left (x\right )^{2}}{4 \, x}\right )} \]
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Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.42 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {x e^{- \frac {- \frac {x^{6}}{4} + 2 x^{5} - 3 x^{4} - 4 x^{3} + \frac {\left (- 8 x^{3} + 32 x^{2} + 16 x\right ) \log {\left (x \right )}}{4} - 4 \log {\left (x \right )}^{2}}{x}}}{5} \]
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Time = 0.36 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {e^{\left (\frac {1}{4} \, x^{5} - 2 \, x^{4} + 3 \, x^{3} + 2 \, x^{2} \log \left (x\right ) + 4 \, x^{2} - 8 \, x \log \left (x\right ) + \frac {4 \, \log \left (x\right )^{2}}{x}\right )}}{5 \, x^{3}} \]
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Time = 0.34 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.47 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {1}{5} \, x e^{\left (\frac {x^{6} - 8 \, x^{5} + 12 \, x^{4} + 8 \, x^{3} \log \left (x\right ) + 16 \, x^{3} - 32 \, x^{2} \log \left (x\right ) - 16 \, x \log \left (x\right ) + 16 \, \log \left (x\right )^{2}}{4 \, x}\right )} \]
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Time = 10.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}} \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx=\frac {x^{2\,x^2-8\,x-3}\,{\mathrm {e}}^{4\,x^2}\,{\mathrm {e}}^{3\,x^3}\,{\mathrm {e}}^{-2\,x^4}\,{\mathrm {e}}^{\frac {x^5}{4}}\,{\mathrm {e}}^{\frac {4\,{\ln \left (x\right )}^2}{x}}}{5} \]
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