Integrand size = 119, antiderivative size = 24 \[ \int \frac {e^{-x} \left (9 x+3 x^2+\left (-9 x^2-3 x^3+\left (-24 x^2+3 x^4\right ) \log (x)+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (3+x)\right ) \log (\log (x))+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (\log (x)) \log (\log (\log (x)))\right )}{(3+x) \log (\log (x))} \, dx=3 e^{-x} x^2 \log (x) (-x+\log (3+x)+\log (\log (\log (x)))) \]
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\[ \int \frac {e^{-x} \left (9 x+3 x^2+\left (-9 x^2-3 x^3+\left (-24 x^2+3 x^4\right ) \log (x)+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (3+x)\right ) \log (\log (x))+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (\log (x)) \log (\log (\log (x)))\right )}{(3+x) \log (\log (x))} \, dx=\int \frac {e^{-x} \left (9 x+3 x^2+\left (-9 x^2-3 x^3+\left (-24 x^2+3 x^4\right ) \log (x)+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (3+x)\right ) \log (\log (x))+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (\log (x)) \log (\log (\log (x)))\right )}{(3+x) \log (\log (x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {3 e^{-x} x \left (3+x+\log (\log (x)) \left (-((3+x) (x-\log (3+x)-\log (\log (\log (x)))))+\log (x) \left (x \left (-8+x^2\right )-\left (-6+x+x^2\right ) \log (3+x)-\left (-6+x+x^2\right ) \log (\log (\log (x)))\right )\right )\right )}{(3+x) \log (\log (x))} \, dx \\ & = 3 \int \frac {e^{-x} x \left (3+x+\log (\log (x)) \left (-((3+x) (x-\log (3+x)-\log (\log (\log (x)))))+\log (x) \left (x \left (-8+x^2\right )-\left (-6+x+x^2\right ) \log (3+x)-\left (-6+x+x^2\right ) \log (\log (\log (x)))\right )\right )\right )}{(3+x) \log (\log (x))} \, dx \\ & = 3 \int \left (\frac {e^{-x} x \left (3+x-3 x \log (\log (x))-x^2 \log (\log (x))-8 x \log (x) \log (\log (x))+x^3 \log (x) \log (\log (x))+3 \log (3+x) \log (\log (x))+x \log (3+x) \log (\log (x))+6 \log (x) \log (3+x) \log (\log (x))-x \log (x) \log (3+x) \log (\log (x))-x^2 \log (x) \log (3+x) \log (\log (x))\right )}{(3+x) \log (\log (x))}-e^{-x} x (-1-2 \log (x)+x \log (x)) \log (\log (\log (x)))\right ) \, dx \\ & = 3 \int \frac {e^{-x} x \left (3+x-3 x \log (\log (x))-x^2 \log (\log (x))-8 x \log (x) \log (\log (x))+x^3 \log (x) \log (\log (x))+3 \log (3+x) \log (\log (x))+x \log (3+x) \log (\log (x))+6 \log (x) \log (3+x) \log (\log (x))-x \log (x) \log (3+x) \log (\log (x))-x^2 \log (x) \log (3+x) \log (\log (x))\right )}{(3+x) \log (\log (x))} \, dx-3 \int e^{-x} x (-1-2 \log (x)+x \log (x)) \log (\log (\log (x))) \, dx \\ & = 3 \int \frac {e^{-x} x \left (3+x+\left (-((3+x) (x-\log (3+x)))+\log (x) \left (x \left (-8+x^2\right )-\left (-6+x+x^2\right ) \log (3+x)\right )\right ) \log (\log (x))\right )}{(3+x) \log (\log (x))} \, dx-3 \int \left (-e^{-x} x \log (\log (\log (x)))-2 e^{-x} x \log (x) \log (\log (\log (x)))+e^{-x} x^2 \log (x) \log (\log (\log (x)))\right ) \, dx \\ & = 3 \int \left (\frac {e^{-x} x \left (-3 x-x^2-8 x \log (x)+x^3 \log (x)+3 \log (3+x)+x \log (3+x)+6 \log (x) \log (3+x)-x \log (x) \log (3+x)-x^2 \log (x) \log (3+x)\right )}{3+x}+\frac {e^{-x} x}{\log (\log (x))}\right ) \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx \\ & = 3 \int \frac {e^{-x} x \left (-3 x-x^2-8 x \log (x)+x^3 \log (x)+3 \log (3+x)+x \log (3+x)+6 \log (x) \log (3+x)-x \log (x) \log (3+x)-x^2 \log (x) \log (3+x)\right )}{3+x} \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx \\ & = 3 \int \frac {e^{-x} x \left (-((3+x) (x-\log (3+x)))+\log (x) \left (x \left (-8+x^2\right )-\left (-6+x+x^2\right ) \log (3+x)\right )\right )}{3+x} \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx \\ & = 3 \int \left (\frac {e^{-x} x^2 \left (-3-x-8 \log (x)+x^2 \log (x)\right )}{3+x}-e^{-x} x (-1-2 \log (x)+x \log (x)) \log (3+x)\right ) \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx \\ & = 3 \int \frac {e^{-x} x^2 \left (-3-x-8 \log (x)+x^2 \log (x)\right )}{3+x} \, dx-3 \int e^{-x} x (-1-2 \log (x)+x \log (x)) \log (3+x) \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx \\ & = 3 \int \left (-e^{-x} x^2+\frac {e^{-x} x^2 \left (-8+x^2\right ) \log (x)}{3+x}\right ) \, dx-3 \int \left (-e^{-x} x \log (3+x)-2 e^{-x} x \log (x) \log (3+x)+e^{-x} x^2 \log (x) \log (3+x)\right ) \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx \\ & = -\left (3 \int e^{-x} x^2 \, dx\right )+3 \int \frac {e^{-x} x^2 \left (-8+x^2\right ) \log (x)}{3+x} \, dx+3 \int e^{-x} x \log (3+x) \, dx-3 \int e^{-x} x^2 \log (x) \log (3+x) \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (3+x) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx \\ & = 3 e^{-x} x^2+6 e^{-x} \log (x)-3 e^{-x} x \log (x)-3 e^{-x} x^3 \log (x)+27 e^3 \text {Ei}(-3-x) \log (x)-3 e^{-x} \log (3+x)-3 e^{-x} x \log (3+x)+3 e^{-x} x^2 \log (x) \log (3+x)-3 \int \frac {e^{-x} (-1-x)}{3+x} \, dx-3 \int \frac {e^{-x} \left (2-x-x^3+9 e^{3+x} \text {Ei}(-3-x)\right )}{x} \, dx+3 \int \frac {e^{-x} \left (-2-2 x-x^2\right ) \log (x)}{3+x} \, dx+3 \int \frac {e^{-x} \left (-2-2 x-x^2\right ) \log (3+x)}{x} \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx-6 \int e^{-x} x \, dx-6 \int \frac {e^{-x} (-1-x) \log (x)}{3+x} \, dx-6 \int \frac {e^{-x} (-1-x) \log (3+x)}{x} \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx \\ & = 6 e^{-x} x+3 e^{-x} x^2-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)-3 \int \left (-e^{-x}+\frac {2 e^{-x}}{3+x}\right ) \, dx-3 \int \left (e^{-x}-\frac {5 e^3 \text {Ei}(-3-x)}{x}\right ) \, dx-3 \int \left (\frac {e^{-x} \left (2-x-x^3\right )}{x}+\frac {9 e^3 \text {Ei}(-3-x)}{x}\right ) \, dx-3 \int \frac {e^{-x} \left (3+x-2 e^x \text {Ei}(-x)\right )}{3+x} \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx-6 \int e^{-x} \, dx+6 \int \frac {e^{-x}+2 e^3 \text {Ei}(-3-x)}{x} \, dx+6 \int \frac {e^{-x}-\text {Ei}(-x)}{3+x} \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx \\ & = 6 e^{-x}+6 e^{-x} x+3 e^{-x} x^2-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)-3 \int \frac {e^{-x} \left (2-x-x^3\right )}{x} \, dx-3 \int \left (e^{-x}-\frac {2 \text {Ei}(-x)}{3+x}\right ) \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx-6 \int \frac {e^{-x}}{3+x} \, dx+6 \int \left (\frac {e^{-x}}{x}+\frac {2 e^3 \text {Ei}(-3-x)}{x}\right ) \, dx+6 \int \left (\frac {e^{-x}}{3+x}-\frac {\text {Ei}(-x)}{3+x}\right ) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx+\left (15 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx-\left (27 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx \\ & = 6 e^{-x}+6 e^{-x} x+3 e^{-x} x^2-6 e^3 \text {Ei}(-3-x)-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)-3 \int e^{-x} \, dx-3 \int \left (-e^{-x}+\frac {2 e^{-x}}{x}-e^{-x} x^2\right ) \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int \frac {e^{-x}}{x} \, dx+6 \int \frac {e^{-x}}{3+x} \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx+\left (12 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx+\left (15 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx-\left (27 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx \\ & = 9 e^{-x}+6 e^{-x} x+3 e^{-x} x^2+6 \text {Ei}(-x)-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)+3 \int e^{-x} \, dx+3 \int e^{-x} x^2 \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx-6 \int \frac {e^{-x}}{x} \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx+\left (12 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx+\left (15 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx-\left (27 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx \\ & = 6 e^{-x}+6 e^{-x} x-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx+\left (12 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx+\left (15 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx-\left (27 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx \\ & = 6 e^{-x}-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx+\left (12 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx+\left (15 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx-\left (27 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx \\ & = -3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx+\left (12 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx+\left (15 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx-\left (27 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx \\ \end{align*}
Time = 5.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-x} \left (9 x+3 x^2+\left (-9 x^2-3 x^3+\left (-24 x^2+3 x^4\right ) \log (x)+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (3+x)\right ) \log (\log (x))+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (\log (x)) \log (\log (\log (x)))\right )}{(3+x) \log (\log (x))} \, dx=-3 e^{-x} x^2 \log (x) (x-\log (3+x)-\log (\log (\log (x)))) \]
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Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50
\[3 x^{2} {\mathrm e}^{-x} \ln \left (x \right ) \ln \left (\ln \left (\ln \left (x \right )\right )\right )-3 x^{2} \ln \left (x \right ) \left (-\ln \left (3+x \right )+x \right ) {\mathrm e}^{-x}\]
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {e^{-x} \left (9 x+3 x^2+\left (-9 x^2-3 x^3+\left (-24 x^2+3 x^4\right ) \log (x)+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (3+x)\right ) \log (\log (x))+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (\log (x)) \log (\log (\log (x)))\right )}{(3+x) \log (\log (x))} \, dx=-3 \, x^{3} e^{\left (-x\right )} \log \left (x\right ) + 3 \, x^{2} e^{\left (-x\right )} \log \left (x + 3\right ) \log \left (x\right ) + 3 \, x^{2} e^{\left (-x\right )} \log \left (x\right ) \log \left (\log \left (\log \left (x\right )\right )\right ) \]
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Timed out. \[ \int \frac {e^{-x} \left (9 x+3 x^2+\left (-9 x^2-3 x^3+\left (-24 x^2+3 x^4\right ) \log (x)+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (3+x)\right ) \log (\log (x))+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (\log (x)) \log (\log (\log (x)))\right )}{(3+x) \log (\log (x))} \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {e^{-x} \left (9 x+3 x^2+\left (-9 x^2-3 x^3+\left (-24 x^2+3 x^4\right ) \log (x)+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (3+x)\right ) \log (\log (x))+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (\log (x)) \log (\log (\log (x)))\right )}{(3+x) \log (\log (x))} \, dx=-3 \, x^{3} e^{\left (-x\right )} \log \left (x\right ) + 3 \, x^{2} e^{\left (-x\right )} \log \left (x + 3\right ) \log \left (x\right ) + 3 \, x^{2} e^{\left (-x\right )} \log \left (x\right ) \log \left (\log \left (\log \left (x\right )\right )\right ) \]
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Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {e^{-x} \left (9 x+3 x^2+\left (-9 x^2-3 x^3+\left (-24 x^2+3 x^4\right ) \log (x)+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (3+x)\right ) \log (\log (x))+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (\log (x)) \log (\log (\log (x)))\right )}{(3+x) \log (\log (x))} \, dx=-3 \, x^{3} e^{\left (-x\right )} \log \left (x\right ) + 3 \, x^{2} e^{\left (-x\right )} \log \left (x + 3\right ) \log \left (x\right ) + 3 \, x^{2} e^{\left (-x\right )} \log \left (x\right ) \log \left (\log \left (\log \left (x\right )\right )\right ) \]
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Time = 12.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-x} \left (9 x+3 x^2+\left (-9 x^2-3 x^3+\left (-24 x^2+3 x^4\right ) \log (x)+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (3+x)\right ) \log (\log (x))+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (\log (x)) \log (\log (\log (x)))\right )}{(3+x) \log (\log (x))} \, dx=3\,x^2\,{\mathrm {e}}^{-x}\,\ln \left (x\right )\,\left (\ln \left (x+3\right )-x+\ln \left (\ln \left (\ln \left (x\right )\right )\right )\right ) \]
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