Integrand size = 137, antiderivative size = 26 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=\left (-x+\frac {24}{4+\frac {e^x}{3 x^2 \log (x)}}\right )^2 \]
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\[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=\int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x \left (e^{2 x} \left (e^x-72 x\right )+108 e^x x \left (e^x (-2+x)-8 (-6+x) x\right ) \log (x)+432 e^x x^2 \left (24-20 x+3 x^2\right ) \log ^2(x)+1728 (-6+x) x^5 \log ^3(x)\right )}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx \\ & = 2 \int \frac {x \left (e^{2 x} \left (e^x-72 x\right )+108 e^x x \left (e^x (-2+x)-8 (-6+x) x\right ) \log (x)+432 e^x x^2 \left (24-20 x+3 x^2\right ) \log ^2(x)+1728 (-6+x) x^5 \log ^3(x)\right )}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx \\ & = 2 \int \left (x+\frac {62208 x^5 \log ^2(x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^3}-\frac {864 x^3 (6+x) \log (x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^2}+\frac {72 x^2 (-1-3 \log (x)+x \log (x))}{e^x+12 x^2 \log (x)}\right ) \, dx \\ & = x^2+144 \int \frac {x^2 (-1-3 \log (x)+x \log (x))}{e^x+12 x^2 \log (x)} \, dx-1728 \int \frac {x^3 (6+x) \log (x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx+124416 \int \frac {x^5 \log ^2(x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx \\ & = x^2+144 \int \left (-\frac {x^2}{e^x+12 x^2 \log (x)}-\frac {3 x^2 \log (x)}{e^x+12 x^2 \log (x)}+\frac {x^3 \log (x)}{e^x+12 x^2 \log (x)}\right ) \, dx-1728 \int \left (\frac {6 x^3 \log (x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^2}+\frac {x^4 \log (x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^2}\right ) \, dx+124416 \int \left (-\frac {x^5 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^3}-\frac {2 x^5 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3}+\frac {x^6 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3}\right ) \, dx \\ & = x^2-144 \int \frac {x^2}{e^x+12 x^2 \log (x)} \, dx+144 \int \frac {x^3 \log (x)}{e^x+12 x^2 \log (x)} \, dx-432 \int \frac {x^2 \log (x)}{e^x+12 x^2 \log (x)} \, dx-1728 \int \frac {x^4 \log (x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx-10368 \int \frac {x^3 \log (x) (-1-2 \log (x)+x \log (x))}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx-124416 \int \frac {x^5 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx+124416 \int \frac {x^6 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx-248832 \int \frac {x^5 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx \\ & = x^2-144 \int \frac {x^2}{e^x+12 x^2 \log (x)} \, dx+144 \int \frac {x^3 \log (x)}{e^x+12 x^2 \log (x)} \, dx-432 \int \frac {x^2 \log (x)}{e^x+12 x^2 \log (x)} \, dx-1728 \int \left (-\frac {x^4 \log (x)}{\left (e^x+12 x^2 \log (x)\right )^2}-\frac {2 x^4 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2}+\frac {x^5 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2}\right ) \, dx-10368 \int \left (-\frac {x^3 \log (x)}{\left (e^x+12 x^2 \log (x)\right )^2}-\frac {2 x^3 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2}+\frac {x^4 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2}\right ) \, dx-124416 \int \frac {x^5 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx+124416 \int \frac {x^6 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx-248832 \int \frac {x^5 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx \\ & = x^2-144 \int \frac {x^2}{e^x+12 x^2 \log (x)} \, dx+144 \int \frac {x^3 \log (x)}{e^x+12 x^2 \log (x)} \, dx-432 \int \frac {x^2 \log (x)}{e^x+12 x^2 \log (x)} \, dx+1728 \int \frac {x^4 \log (x)}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx-1728 \int \frac {x^5 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx+3456 \int \frac {x^4 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx+10368 \int \frac {x^3 \log (x)}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx-10368 \int \frac {x^4 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx+20736 \int \frac {x^3 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^2} \, dx-124416 \int \frac {x^5 \log ^2(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx+124416 \int \frac {x^6 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx-248832 \int \frac {x^5 \log ^3(x)}{\left (e^x+12 x^2 \log (x)\right )^3} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(26)=52\).
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=2 \left (-6 x+\frac {x^2}{2}+\frac {18 e^{2 x}}{\left (e^x+12 x^2 \log (x)\right )^2}+\frac {6 e^x (-6+x)}{e^x+12 x^2 \log (x)}\right ) \]
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Time = 2.38 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81
method | result | size |
risch | \(x^{2}-12 x +\frac {12 \left (12 x^{3} \ln \left (x \right )-72 x^{2} \ln \left (x \right )+{\mathrm e}^{x} x -3 \,{\mathrm e}^{x}\right ) {\mathrm e}^{x}}{\left (12 x^{2} \ln \left (x \right )+{\mathrm e}^{x}\right )^{2}}\) | \(47\) |
parallelrisch | \(-\frac {-3456 x^{6} \ln \left (x \right )^{2}+41472 x^{5} \ln \left (x \right )^{2}-124416 x^{4} \ln \left (x \right )^{2}-576 x^{4} {\mathrm e}^{x} \ln \left (x \right )+3456 x^{3} {\mathrm e}^{x} \ln \left (x \right )-24 \,{\mathrm e}^{2 x} x^{2}}{24 \left (144 x^{4} \ln \left (x \right )^{2}+24 x^{2} {\mathrm e}^{x} \ln \left (x \right )+{\mathrm e}^{2 x}\right )}\) | \(83\) |
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (20) = 40\).
Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.77 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=\frac {24 \, {\left (x^{4} - 6 \, x^{3} - 36 \, x^{2}\right )} e^{x} \log \left (x\right ) + 144 \, {\left (x^{6} - 12 \, x^{5}\right )} \log \left (x\right )^{2} + {\left (x^{2} - 36\right )} e^{\left (2 \, x\right )}}{144 \, x^{4} \log \left (x\right )^{2} + 24 \, x^{2} e^{x} \log \left (x\right ) + e^{\left (2 \, x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=x^{2} + \frac {- 1728 x^{5} \log {\left (x \right )}^{2} + 5184 x^{4} \log {\left (x \right )}^{2} - 144 x^{3} e^{x} \log {\left (x \right )}}{144 x^{4} \log {\left (x \right )}^{2} + 24 x^{2} e^{x} \log {\left (x \right )} + e^{2 x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=\frac {x^{2} e^{\left (2 \, x\right )} + 24 \, {\left (x^{4} - 6 \, x^{3}\right )} e^{x} \log \left (x\right ) + 144 \, {\left (x^{6} - 12 \, x^{5} + 36 \, x^{4}\right )} \log \left (x\right )^{2}}{144 \, x^{4} \log \left (x\right )^{2} + 24 \, x^{2} e^{x} \log \left (x\right ) + e^{\left (2 \, x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (20) = 40\).
Time = 0.40 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.65 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=\frac {144 \, x^{6} \log \left (x\right )^{2} - 1728 \, x^{5} \log \left (x\right )^{2} + 24 \, x^{4} e^{x} \log \left (x\right ) - 1872 \, x^{4} \log \left (x\right )^{2} - 144 \, x^{3} e^{x} \log \left (x\right ) - 1176 \, x^{2} e^{x} \log \left (x\right ) + x^{2} e^{\left (2 \, x\right )} - 49 \, e^{\left (2 \, x\right )}}{144 \, x^{4} \log \left (x\right )^{2} + 24 \, x^{2} e^{x} \log \left (x\right ) + e^{\left (2 \, x\right )}} \]
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Time = 9.01 (sec) , antiderivative size = 155, normalized size of antiderivative = 5.96 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=x^2-12\,x+\frac {12\,\left (12\,x\,{\mathrm {e}}^{2\,x}-72\,x^3\,{\mathrm {e}}^x+12\,x^4\,{\mathrm {e}}^x-8\,x^2\,{\mathrm {e}}^{2\,x}+x^3\,{\mathrm {e}}^{2\,x}\right )}{\left ({\mathrm {e}}^x+12\,x^2\,\ln \left (x\right )\right )\,\left (x^2\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^x+12\,x^3\right )}+\frac {36\,{\mathrm {e}}^x\,\left (12\,x^5\,{\mathrm {e}}^x-2\,x^3\,{\mathrm {e}}^{2\,x}+x^4\,{\mathrm {e}}^{2\,x}\right )}{x^2\,\left ({\mathrm {e}}^{2\,x}+144\,x^4\,{\ln \left (x\right )}^2+24\,x^2\,{\mathrm {e}}^x\,\ln \left (x\right )\right )\,\left (x^2\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^x+12\,x^3\right )} \]
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