Integrand size = 106, antiderivative size = 24 \[ \int \frac {e^{-8-4 x} \left (-64-32 x-4 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)+\left (-96-48 x-6 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)\right ) \log (\log (x))+\left (-32-16 x-2 x^2+\left (-56 x-30 x^2-4 x^3\right ) \log (x)\right ) \log ^2(\log (x))\right )}{x \log ^3(x)} \, dx=\frac {e^{-8-4 x} (4+x)^2 (2+\log (\log (x)))^2}{\log ^2(x)} \]
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\[ \int \frac {e^{-8-4 x} \left (-64-32 x-4 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)+\left (-96-48 x-6 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)\right ) \log (\log (x))+\left (-32-16 x-2 x^2+\left (-56 x-30 x^2-4 x^3\right ) \log (x)\right ) \log ^2(\log (x))\right )}{x \log ^3(x)} \, dx=\int \frac {e^{-8-4 x} \left (-64-32 x-4 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)+\left (-96-48 x-6 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)\right ) \log (\log (x))+\left (-32-16 x-2 x^2+\left (-56 x-30 x^2-4 x^3\right ) \log (x)\right ) \log ^2(\log (x))\right )}{x \log ^3(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e^{-8-4 x} (4+x) (2+\log (\log (x))) (-((4+x) (1+\log (\log (x))))-x (7+2 x) \log (x) (2+\log (\log (x))))}{x \log ^3(x)} \, dx \\ & = 2 \int \frac {e^{-8-4 x} (4+x) (2+\log (\log (x))) (-((4+x) (1+\log (\log (x))))-x (7+2 x) \log (x) (2+\log (\log (x))))}{x \log ^3(x)} \, dx \\ & = 2 \int \left (-\frac {2 e^{-8-4 x} (4+x) \left (4+x+14 x \log (x)+4 x^2 \log (x)\right )}{x \log ^3(x)}-\frac {e^{-8-4 x} (4+x) \left (12+3 x+28 x \log (x)+8 x^2 \log (x)\right ) \log (\log (x))}{x \log ^3(x)}-\frac {e^{-8-4 x} (4+x) \left (4+x+7 x \log (x)+2 x^2 \log (x)\right ) \log ^2(\log (x))}{x \log ^3(x)}\right ) \, dx \\ & = -\left (2 \int \frac {e^{-8-4 x} (4+x) \left (12+3 x+28 x \log (x)+8 x^2 \log (x)\right ) \log (\log (x))}{x \log ^3(x)} \, dx\right )-2 \int \frac {e^{-8-4 x} (4+x) \left (4+x+7 x \log (x)+2 x^2 \log (x)\right ) \log ^2(\log (x))}{x \log ^3(x)} \, dx-4 \int \frac {e^{-8-4 x} (4+x) \left (4+x+14 x \log (x)+4 x^2 \log (x)\right )}{x \log ^3(x)} \, dx \\ & = -\left (2 \int \left (\frac {e^{-8-4 x} \left (12+3 x+28 x \log (x)+8 x^2 \log (x)\right ) \log (\log (x))}{\log ^3(x)}+\frac {4 e^{-8-4 x} \left (12+3 x+28 x \log (x)+8 x^2 \log (x)\right ) \log (\log (x))}{x \log ^3(x)}\right ) \, dx\right )-2 \int \left (\frac {e^{-8-4 x} \left (4+x+7 x \log (x)+2 x^2 \log (x)\right ) \log ^2(\log (x))}{\log ^3(x)}+\frac {4 e^{-8-4 x} \left (4+x+7 x \log (x)+2 x^2 \log (x)\right ) \log ^2(\log (x))}{x \log ^3(x)}\right ) \, dx-4 \int \left (\frac {e^{-8-4 x} (4+x)^2}{x \log ^3(x)}+\frac {2 e^{-8-4 x} \left (28+15 x+2 x^2\right )}{\log ^2(x)}\right ) \, dx \\ & = -\left (2 \int \frac {e^{-8-4 x} \left (12+3 x+28 x \log (x)+8 x^2 \log (x)\right ) \log (\log (x))}{\log ^3(x)} \, dx\right )-2 \int \frac {e^{-8-4 x} \left (4+x+7 x \log (x)+2 x^2 \log (x)\right ) \log ^2(\log (x))}{\log ^3(x)} \, dx-4 \int \frac {e^{-8-4 x} (4+x)^2}{x \log ^3(x)} \, dx-8 \int \frac {e^{-8-4 x} \left (28+15 x+2 x^2\right )}{\log ^2(x)} \, dx-8 \int \frac {e^{-8-4 x} \left (12+3 x+28 x \log (x)+8 x^2 \log (x)\right ) \log (\log (x))}{x \log ^3(x)} \, dx-8 \int \frac {e^{-8-4 x} \left (4+x+7 x \log (x)+2 x^2 \log (x)\right ) \log ^2(\log (x))}{x \log ^3(x)} \, dx \\ & = -\left (2 \int \left (\frac {12 e^{-8-4 x} \log (\log (x))}{\log ^3(x)}+\frac {3 e^{-8-4 x} x \log (\log (x))}{\log ^3(x)}+\frac {28 e^{-8-4 x} x \log (\log (x))}{\log ^2(x)}+\frac {8 e^{-8-4 x} x^2 \log (\log (x))}{\log ^2(x)}\right ) \, dx\right )-2 \int \left (\frac {4 e^{-8-4 x} \log ^2(\log (x))}{\log ^3(x)}+\frac {e^{-8-4 x} x \log ^2(\log (x))}{\log ^3(x)}+\frac {7 e^{-8-4 x} x \log ^2(\log (x))}{\log ^2(x)}+\frac {2 e^{-8-4 x} x^2 \log ^2(\log (x))}{\log ^2(x)}\right ) \, dx-4 \int \left (\frac {8 e^{-8-4 x}}{\log ^3(x)}+\frac {16 e^{-8-4 x}}{x \log ^3(x)}+\frac {e^{-8-4 x} x}{\log ^3(x)}\right ) \, dx-8 \int \left (\frac {28 e^{-8-4 x}}{\log ^2(x)}+\frac {15 e^{-8-4 x} x}{\log ^2(x)}+\frac {2 e^{-8-4 x} x^2}{\log ^2(x)}\right ) \, dx-8 \int \left (\frac {3 e^{-8-4 x} \log (\log (x))}{\log ^3(x)}+\frac {12 e^{-8-4 x} \log (\log (x))}{x \log ^3(x)}+\frac {28 e^{-8-4 x} \log (\log (x))}{\log ^2(x)}+\frac {8 e^{-8-4 x} x \log (\log (x))}{\log ^2(x)}\right ) \, dx-8 \int \left (\frac {e^{-8-4 x} \log ^2(\log (x))}{\log ^3(x)}+\frac {4 e^{-8-4 x} \log ^2(\log (x))}{x \log ^3(x)}+\frac {7 e^{-8-4 x} \log ^2(\log (x))}{\log ^2(x)}+\frac {2 e^{-8-4 x} x \log ^2(\log (x))}{\log ^2(x)}\right ) \, dx \\ & = -\left (2 \int \frac {e^{-8-4 x} x \log ^2(\log (x))}{\log ^3(x)} \, dx\right )-4 \int \frac {e^{-8-4 x} x}{\log ^3(x)} \, dx-4 \int \frac {e^{-8-4 x} x^2 \log ^2(\log (x))}{\log ^2(x)} \, dx-6 \int \frac {e^{-8-4 x} x \log (\log (x))}{\log ^3(x)} \, dx-2 \left (8 \int \frac {e^{-8-4 x} \log ^2(\log (x))}{\log ^3(x)} \, dx\right )-14 \int \frac {e^{-8-4 x} x \log ^2(\log (x))}{\log ^2(x)} \, dx-16 \int \frac {e^{-8-4 x} x^2}{\log ^2(x)} \, dx-16 \int \frac {e^{-8-4 x} x^2 \log (\log (x))}{\log ^2(x)} \, dx-16 \int \frac {e^{-8-4 x} x \log ^2(\log (x))}{\log ^2(x)} \, dx-2 \left (24 \int \frac {e^{-8-4 x} \log (\log (x))}{\log ^3(x)} \, dx\right )-32 \int \frac {e^{-8-4 x}}{\log ^3(x)} \, dx-32 \int \frac {e^{-8-4 x} \log ^2(\log (x))}{x \log ^3(x)} \, dx-56 \int \frac {e^{-8-4 x} x \log (\log (x))}{\log ^2(x)} \, dx-56 \int \frac {e^{-8-4 x} \log ^2(\log (x))}{\log ^2(x)} \, dx-64 \int \frac {e^{-8-4 x}}{x \log ^3(x)} \, dx-64 \int \frac {e^{-8-4 x} x \log (\log (x))}{\log ^2(x)} \, dx-96 \int \frac {e^{-8-4 x} \log (\log (x))}{x \log ^3(x)} \, dx-120 \int \frac {e^{-8-4 x} x}{\log ^2(x)} \, dx-224 \int \frac {e^{-8-4 x}}{\log ^2(x)} \, dx-224 \int \frac {e^{-8-4 x} \log (\log (x))}{\log ^2(x)} \, dx \\ \end{align*}
Time = 2.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-8-4 x} \left (-64-32 x-4 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)+\left (-96-48 x-6 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)\right ) \log (\log (x))+\left (-32-16 x-2 x^2+\left (-56 x-30 x^2-4 x^3\right ) \log (x)\right ) \log ^2(\log (x))\right )}{x \log ^3(x)} \, dx=\frac {e^{-4 (2+x)} (4+x)^2 (2+\log (\log (x)))^2}{\log ^2(x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(68\) vs. \(2(25)=50\).
Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.88
method | result | size |
risch | \(\frac {\left (x^{2}+8 x +16\right ) {\mathrm e}^{-4 x -8} \ln \left (\ln \left (x \right )\right )^{2}}{\ln \left (x \right )^{2}}+\frac {4 \left (x^{2}+8 x +16\right ) {\mathrm e}^{-4 x -8} \ln \left (\ln \left (x \right )\right )}{\ln \left (x \right )^{2}}+\frac {4 \left (x^{2}+8 x +16\right ) {\mathrm e}^{-4 x -8}}{\ln \left (x \right )^{2}}\) | \(69\) |
parallelrisch | \(-\frac {\left (-128-2 x^{2} \ln \left (\ln \left (x \right )\right )^{2}-8 x^{2} \ln \left (\ln \left (x \right )\right )-16 x \ln \left (\ln \left (x \right )\right )^{2}-8 x^{2}-64 x \ln \left (\ln \left (x \right )\right )-32 \ln \left (\ln \left (x \right )\right )^{2}-64 x -128 \ln \left (\ln \left (x \right )\right )\right ) {\mathrm e}^{-4 x -8}}{2 \ln \left (x \right )^{2}}\) | \(69\) |
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.54 \[ \int \frac {e^{-8-4 x} \left (-64-32 x-4 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)+\left (-96-48 x-6 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)\right ) \log (\log (x))+\left (-32-16 x-2 x^2+\left (-56 x-30 x^2-4 x^3\right ) \log (x)\right ) \log ^2(\log (x))\right )}{x \log ^3(x)} \, dx=\frac {{\left (x^{2} + 8 \, x + 16\right )} e^{\left (-4 \, x - 8\right )} \log \left (\log \left (x\right )\right )^{2} + 4 \, {\left (x^{2} + 8 \, x + 16\right )} e^{\left (-4 \, x - 8\right )} \log \left (\log \left (x\right )\right ) + 4 \, {\left (x^{2} + 8 \, x + 16\right )} e^{\left (-4 \, x - 8\right )}}{\log \left (x\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (26) = 52\).
Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.17 \[ \int \frac {e^{-8-4 x} \left (-64-32 x-4 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)+\left (-96-48 x-6 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)\right ) \log (\log (x))+\left (-32-16 x-2 x^2+\left (-56 x-30 x^2-4 x^3\right ) \log (x)\right ) \log ^2(\log (x))\right )}{x \log ^3(x)} \, dx=\frac {\left (x^{2} \log {\left (\log {\left (x \right )} \right )}^{2} + 4 x^{2} \log {\left (\log {\left (x \right )} \right )} + 4 x^{2} + 8 x \log {\left (\log {\left (x \right )} \right )}^{2} + 32 x \log {\left (\log {\left (x \right )} \right )} + 32 x + 16 \log {\left (\log {\left (x \right )} \right )}^{2} + 64 \log {\left (\log {\left (x \right )} \right )} + 64\right ) e^{- 4 x - 8}}{\log {\left (x \right )}^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {e^{-8-4 x} \left (-64-32 x-4 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)+\left (-96-48 x-6 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)\right ) \log (\log (x))+\left (-32-16 x-2 x^2+\left (-56 x-30 x^2-4 x^3\right ) \log (x)\right ) \log ^2(\log (x))\right )}{x \log ^3(x)} \, dx=\frac {{\left ({\left (x^{2} + 8 \, x + 16\right )} e^{\left (-4 \, x\right )} \log \left (\log \left (x\right )\right )^{2} + 4 \, {\left (x^{2} + 8 \, x + 16\right )} e^{\left (-4 \, x\right )} \log \left (\log \left (x\right )\right ) + 4 \, {\left (x^{2} + 8 \, x + 16\right )} e^{\left (-4 \, x\right )}\right )} e^{\left (-8\right )}}{\log \left (x\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (23) = 46\).
Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 4.04 \[ \int \frac {e^{-8-4 x} \left (-64-32 x-4 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)+\left (-96-48 x-6 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)\right ) \log (\log (x))+\left (-32-16 x-2 x^2+\left (-56 x-30 x^2-4 x^3\right ) \log (x)\right ) \log ^2(\log (x))\right )}{x \log ^3(x)} \, dx=\frac {{\left (x^{2} e^{\left (-4 \, x\right )} \log \left (\log \left (x\right )\right )^{2} + 4 \, x^{2} e^{\left (-4 \, x\right )} \log \left (\log \left (x\right )\right ) + 8 \, x e^{\left (-4 \, x\right )} \log \left (\log \left (x\right )\right )^{2} + 4 \, x^{2} e^{\left (-4 \, x\right )} + 32 \, x e^{\left (-4 \, x\right )} \log \left (\log \left (x\right )\right ) + 16 \, e^{\left (-4 \, x\right )} \log \left (\log \left (x\right )\right )^{2} + 32 \, x e^{\left (-4 \, x\right )} + 64 \, e^{\left (-4 \, x\right )} \log \left (\log \left (x\right )\right ) + 64 \, e^{\left (-4 \, x\right )}\right )} e^{\left (-8\right )}}{\log \left (x\right )^{2}} \]
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Timed out. \[ \int \frac {e^{-8-4 x} \left (-64-32 x-4 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)+\left (-96-48 x-6 x^2+\left (-224 x-120 x^2-16 x^3\right ) \log (x)\right ) \log (\log (x))+\left (-32-16 x-2 x^2+\left (-56 x-30 x^2-4 x^3\right ) \log (x)\right ) \log ^2(\log (x))\right )}{x \log ^3(x)} \, dx=\int -\frac {{\mathrm {e}}^{-4\,x-8}\,\left (32\,x+\ln \left (\ln \left (x\right )\right )\,\left (48\,x+6\,x^2+\ln \left (x\right )\,\left (16\,x^3+120\,x^2+224\,x\right )+96\right )+{\ln \left (\ln \left (x\right )\right )}^2\,\left (16\,x+2\,x^2+\ln \left (x\right )\,\left (4\,x^3+30\,x^2+56\,x\right )+32\right )+4\,x^2+\ln \left (x\right )\,\left (16\,x^3+120\,x^2+224\,x\right )+64\right )}{x\,{\ln \left (x\right )}^3} \,d x \]
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