Integrand size = 89, antiderivative size = 27 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{-x+\frac {x \left (5+x+(4-2 x-\log (x))^2\right )}{-3+x}} \]
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\[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=\int \frac {\exp \left (-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}\right ) \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}\right ) \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{(-3+x)^2} \, dx \\ & = \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{(3-x)^2} \, dx \\ & = \int \left (-\frac {48 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2}+\frac {76 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x}{(-3+x)^2}-\frac {48 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x^2}{(-3+x)^2}+\frac {8 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x^3}{(-3+x)^2}+\frac {2 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) (-1+x) (-9+2 x) \log (x)}{(-3+x)^2}-\frac {3 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log ^2(x)}{(-3+x)^2}\right ) \, dx \\ & = 2 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) (-1+x) (-9+2 x) \log (x)}{(-3+x)^2} \, dx-3 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log ^2(x)}{(-3+x)^2} \, dx+8 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x^3}{(-3+x)^2} \, dx-48 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2} \, dx-48 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x^2}{(-3+x)^2} \, dx+76 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x}{(-3+x)^2} \, dx \\ & = 2 \int \left (2 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log (x)-\frac {6 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log (x)}{(-3+x)^2}+\frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log (x)}{-3+x}\right ) \, dx-3 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log ^2(x)}{(-3+x)^2} \, dx+8 \int \left (6 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )+\frac {27 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2}+\frac {27 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{-3+x}+\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x\right ) \, dx-48 \int \left (\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )+\frac {9 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2}+\frac {6 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{-3+x}\right ) \, dx-48 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2} \, dx+76 \int \left (\frac {3 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2}+\frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{-3+x}\right ) \, dx \\ & = 2 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log (x)}{-3+x} \, dx-3 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log ^2(x)}{(-3+x)^2} \, dx+4 \int \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log (x) \, dx+8 \int \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x \, dx-12 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log (x)}{(-3+x)^2} \, dx-48 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2} \, dx+76 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{-3+x} \, dx+216 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2} \, dx+216 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{-3+x} \, dx+228 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2} \, dx-288 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{-3+x} \, dx-432 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2} \, dx \\ \end{align*}
Time = 5.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{\frac {x \left (4 \left (6-4 x+x^2\right )+\log ^2(x)\right )}{-3+x}} x^{\frac {4 (-2+x) x}{-3+x}} \]
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Time = 1.60 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19
method | result | size |
risch | \({\mathrm e}^{\frac {x \left (\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}-8 \ln \left (x \right )-16 x +24\right )}{-3+x}}\) | \(32\) |
parallelrisch | \({\mathrm e}^{-x} {\mathrm e}^{\frac {x \left (\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}-8 \ln \left (x \right )-15 x +21\right )}{-3+x}}\) | \(37\) |
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{\left (\frac {4 \, x^{3} + x \log \left (x\right )^{2} - 16 \, x^{2} + 4 \, {\left (x^{2} - 2 \, x\right )} \log \left (x\right ) + 24 \, x}{x - 3}\right )} \]
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Time = 35.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{- x} e^{\frac {4 x^{3} - 15 x^{2} + x \log {\left (x \right )}^{2} + 21 x + \left (4 x^{2} - 8 x\right ) \log {\left (x \right )}}{x - 3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.35 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=x^{4} e^{\left (4 \, x^{2} + 4 \, x \log \left (x\right ) + \log \left (x\right )^{2} - 4 \, x + \frac {3 \, \log \left (x\right )^{2}}{x - 3} + \frac {12 \, \log \left (x\right )}{x - 3} + \frac {36}{x - 3} + 12\right )} \]
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Time = 0.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{\left (\frac {4 \, x^{3} + 4 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} - 16 \, x^{2} - 8 \, x \log \left (x\right ) + 24 \, x}{x - 3}\right )} \]
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Time = 13.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.52 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{\frac {21\,x}{x-3}}\,{\mathrm {e}}^{\frac {x\,{\ln \left (x\right )}^2}{x-3}}\,{\mathrm {e}}^{\frac {4\,x^3}{x-3}}\,{\mathrm {e}}^{-\frac {15\,x^2}{x-3}}}{x^{\frac {4\,\left (2\,x-x^2\right )}{x-3}}} \]
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