Integrand size = 349, antiderivative size = 31 \[ \int \frac {24 e^2 x^4-e^{2+x} x^6-108 e^2 x^3 \log (x)+\left (-180 e^2 x^3+12 e^{2+x} x^5\right ) \log ^2(x)+324 e^2 x^2 \log ^3(x)+\left (729 e^2 x^2-54 e^{2+x} x^4\right ) \log ^4(x)+\left (-1620 e^2 x+108 e^{2+x} x^3\right ) \log ^6(x)+\left (1215 e^2-81 e^{2+x} x^2\right ) \log ^8(x)}{576 x^4+240 x^5+25 x^6+e^{2 x} x^6+e^x \left (48 x^5+10 x^6\right )+\left (-4320 x^3-2340 x^4-300 x^5-12 e^{2 x} x^5+e^x \left (-468 x^4-120 x^5\right )\right ) \log ^2(x)+\left (14580 x^2+8910 x^3+1350 x^4+54 e^{2 x} x^4+e^x \left (1782 x^3+540 x^4\right )\right ) \log ^4(x)+\left (-24300 x-16200 x^2-2700 x^3-108 e^{2 x} x^3+e^x \left (-3240 x^2-1080 x^3\right )\right ) \log ^6(x)+\left (18225+12150 x+2025 x^2+81 e^{2 x} x^2+e^x \left (2430 x+810 x^2\right )\right ) \log ^8(x)} \, dx=\frac {e^2}{5+e^x+\frac {15}{x}+\frac {9 x}{\left (-x+3 \log ^2(x)\right )^2}} \]
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\[ \int \frac {24 e^2 x^4-e^{2+x} x^6-108 e^2 x^3 \log (x)+\left (-180 e^2 x^3+12 e^{2+x} x^5\right ) \log ^2(x)+324 e^2 x^2 \log ^3(x)+\left (729 e^2 x^2-54 e^{2+x} x^4\right ) \log ^4(x)+\left (-1620 e^2 x+108 e^{2+x} x^3\right ) \log ^6(x)+\left (1215 e^2-81 e^{2+x} x^2\right ) \log ^8(x)}{576 x^4+240 x^5+25 x^6+e^{2 x} x^6+e^x \left (48 x^5+10 x^6\right )+\left (-4320 x^3-2340 x^4-300 x^5-12 e^{2 x} x^5+e^x \left (-468 x^4-120 x^5\right )\right ) \log ^2(x)+\left (14580 x^2+8910 x^3+1350 x^4+54 e^{2 x} x^4+e^x \left (1782 x^3+540 x^4\right )\right ) \log ^4(x)+\left (-24300 x-16200 x^2-2700 x^3-108 e^{2 x} x^3+e^x \left (-3240 x^2-1080 x^3\right )\right ) \log ^6(x)+\left (18225+12150 x+2025 x^2+81 e^{2 x} x^2+e^x \left (2430 x+810 x^2\right )\right ) \log ^8(x)} \, dx=\int \frac {24 e^2 x^4-e^{2+x} x^6-108 e^2 x^3 \log (x)+\left (-180 e^2 x^3+12 e^{2+x} x^5\right ) \log ^2(x)+324 e^2 x^2 \log ^3(x)+\left (729 e^2 x^2-54 e^{2+x} x^4\right ) \log ^4(x)+\left (-1620 e^2 x+108 e^{2+x} x^3\right ) \log ^6(x)+\left (1215 e^2-81 e^{2+x} x^2\right ) \log ^8(x)}{576 x^4+240 x^5+25 x^6+e^{2 x} x^6+e^x \left (48 x^5+10 x^6\right )+\left (-4320 x^3-2340 x^4-300 x^5-12 e^{2 x} x^5+e^x \left (-468 x^4-120 x^5\right )\right ) \log ^2(x)+\left (14580 x^2+8910 x^3+1350 x^4+54 e^{2 x} x^4+e^x \left (1782 x^3+540 x^4\right )\right ) \log ^4(x)+\left (-24300 x-16200 x^2-2700 x^3-108 e^{2 x} x^3+e^x \left (-3240 x^2-1080 x^3\right )\right ) \log ^6(x)+\left (18225+12150 x+2025 x^2+81 e^{2 x} x^2+e^x \left (2430 x+810 x^2\right )\right ) \log ^8(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^2 \left (x-3 \log ^2(x)\right ) \left (-x^3 \left (-24+e^x x^2\right )-108 x^2 \log (x)+9 x^2 \left (-12+e^x x^2\right ) \log ^2(x)-27 x \left (-15+e^x x^2\right ) \log ^4(x)+27 \left (-15+e^x x^2\right ) \log ^6(x)\right )}{\left (x^2 \left (24+\left (5+e^x\right ) x\right )-6 x \left (15+\left (5+e^x\right ) x\right ) \log ^2(x)+9 \left (15+\left (5+e^x\right ) x\right ) \log ^4(x)\right )^2} \, dx \\ & = e^2 \int \frac {\left (x-3 \log ^2(x)\right ) \left (-x^3 \left (-24+e^x x^2\right )-108 x^2 \log (x)+9 x^2 \left (-12+e^x x^2\right ) \log ^2(x)-27 x \left (-15+e^x x^2\right ) \log ^4(x)+27 \left (-15+e^x x^2\right ) \log ^6(x)\right )}{\left (x^2 \left (24+\left (5+e^x\right ) x\right )-6 x \left (15+\left (5+e^x\right ) x\right ) \log ^2(x)+9 \left (15+\left (5+e^x\right ) x\right ) \log ^4(x)\right )^2} \, dx \\ & = e^2 \int \left (-\frac {x \left (x-3 \log ^2(x)\right )^2}{24 x^2+5 x^3+e^x x^3-90 x \log ^2(x)-30 x^2 \log ^2(x)-6 e^x x^2 \log ^2(x)+135 \log ^4(x)+45 x \log ^4(x)+9 e^x x \log ^4(x)}+\frac {24 x^4+24 x^5+5 x^6-108 x^3 \log (x)-180 x^3 \log ^2(x)-234 x^4 \log ^2(x)-60 x^5 \log ^2(x)+324 x^2 \log ^3(x)+729 x^2 \log ^4(x)+891 x^3 \log ^4(x)+270 x^4 \log ^4(x)-1620 x \log ^6(x)-1620 x^2 \log ^6(x)-540 x^3 \log ^6(x)+1215 \log ^8(x)+1215 x \log ^8(x)+405 x^2 \log ^8(x)}{\left (24 x^2+5 x^3+e^x x^3-90 x \log ^2(x)-30 x^2 \log ^2(x)-6 e^x x^2 \log ^2(x)+135 \log ^4(x)+45 x \log ^4(x)+9 e^x x \log ^4(x)\right )^2}\right ) \, dx \\ & = -\left (e^2 \int \frac {x \left (x-3 \log ^2(x)\right )^2}{24 x^2+5 x^3+e^x x^3-90 x \log ^2(x)-30 x^2 \log ^2(x)-6 e^x x^2 \log ^2(x)+135 \log ^4(x)+45 x \log ^4(x)+9 e^x x \log ^4(x)} \, dx\right )+e^2 \int \frac {24 x^4+24 x^5+5 x^6-108 x^3 \log (x)-180 x^3 \log ^2(x)-234 x^4 \log ^2(x)-60 x^5 \log ^2(x)+324 x^2 \log ^3(x)+729 x^2 \log ^4(x)+891 x^3 \log ^4(x)+270 x^4 \log ^4(x)-1620 x \log ^6(x)-1620 x^2 \log ^6(x)-540 x^3 \log ^6(x)+1215 \log ^8(x)+1215 x \log ^8(x)+405 x^2 \log ^8(x)}{\left (24 x^2+5 x^3+e^x x^3-90 x \log ^2(x)-30 x^2 \log ^2(x)-6 e^x x^2 \log ^2(x)+135 \log ^4(x)+45 x \log ^4(x)+9 e^x x \log ^4(x)\right )^2} \, dx \\ & = -\left (e^2 \int \frac {x \left (x-3 \log ^2(x)\right )^2}{x^2 \left (24+\left (5+e^x\right ) x\right )-6 x \left (15+\left (5+e^x\right ) x\right ) \log ^2(x)+9 \left (15+\left (5+e^x\right ) x\right ) \log ^4(x)} \, dx\right )+e^2 \int \frac {x^4 \left (24+24 x+5 x^2\right )-108 x^3 \log (x)-6 x^3 \left (30+39 x+10 x^2\right ) \log ^2(x)+324 x^2 \log ^3(x)+27 x^2 \left (27+33 x+10 x^2\right ) \log ^4(x)-540 x \left (3+3 x+x^2\right ) \log ^6(x)+405 \left (3+3 x+x^2\right ) \log ^8(x)}{\left (x^2 \left (24+\left (5+e^x\right ) x\right )-6 x \left (15+\left (5+e^x\right ) x\right ) \log ^2(x)+9 \left (15+\left (5+e^x\right ) x\right ) \log ^4(x)\right )^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.00 \[ \int \frac {24 e^2 x^4-e^{2+x} x^6-108 e^2 x^3 \log (x)+\left (-180 e^2 x^3+12 e^{2+x} x^5\right ) \log ^2(x)+324 e^2 x^2 \log ^3(x)+\left (729 e^2 x^2-54 e^{2+x} x^4\right ) \log ^4(x)+\left (-1620 e^2 x+108 e^{2+x} x^3\right ) \log ^6(x)+\left (1215 e^2-81 e^{2+x} x^2\right ) \log ^8(x)}{576 x^4+240 x^5+25 x^6+e^{2 x} x^6+e^x \left (48 x^5+10 x^6\right )+\left (-4320 x^3-2340 x^4-300 x^5-12 e^{2 x} x^5+e^x \left (-468 x^4-120 x^5\right )\right ) \log ^2(x)+\left (14580 x^2+8910 x^3+1350 x^4+54 e^{2 x} x^4+e^x \left (1782 x^3+540 x^4\right )\right ) \log ^4(x)+\left (-24300 x-16200 x^2-2700 x^3-108 e^{2 x} x^3+e^x \left (-3240 x^2-1080 x^3\right )\right ) \log ^6(x)+\left (18225+12150 x+2025 x^2+81 e^{2 x} x^2+e^x \left (2430 x+810 x^2\right )\right ) \log ^8(x)} \, dx=\frac {e^2 x \left (x-3 \log ^2(x)\right )^2}{x^2 \left (24+\left (5+e^x\right ) x\right )-6 x \left (15+\left (5+e^x\right ) x\right ) \log ^2(x)+9 \left (15+\left (5+e^x\right ) x\right ) \log ^4(x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs. \(2(29)=58\).
Time = 24.59 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.13
method | result | size |
parallelrisch | \(\frac {-6 x^{2} \ln \left (x \right )^{2} {\mathrm e}^{2}+x^{3} {\mathrm e}^{2}+9 x \ln \left (x \right )^{4} {\mathrm e}^{2}}{9 x \ln \left (x \right )^{4} {\mathrm e}^{x}+45 x \ln \left (x \right )^{4}-6 x^{2} {\mathrm e}^{x} \ln \left (x \right )^{2}+135 \ln \left (x \right )^{4}-30 x^{2} \ln \left (x \right )^{2}+{\mathrm e}^{x} x^{3}-90 x \ln \left (x \right )^{2}+5 x^{3}+24 x^{2}}\) | \(97\) |
risch | \(\frac {{\mathrm e}^{2} x}{{\mathrm e}^{x} x +5 x +15}-\frac {9 x^{3} {\mathrm e}^{2}}{\left ({\mathrm e}^{x} x +5 x +15\right ) \left (9 x \ln \left (x \right )^{4} {\mathrm e}^{x}+45 x \ln \left (x \right )^{4}-6 x^{2} {\mathrm e}^{x} \ln \left (x \right )^{2}+135 \ln \left (x \right )^{4}-30 x^{2} \ln \left (x \right )^{2}+{\mathrm e}^{x} x^{3}-90 x \ln \left (x \right )^{2}+5 x^{3}+24 x^{2}\right )}\) | \(103\) |
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Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (29) = 58\).
Time = 0.32 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.19 \[ \int \frac {24 e^2 x^4-e^{2+x} x^6-108 e^2 x^3 \log (x)+\left (-180 e^2 x^3+12 e^{2+x} x^5\right ) \log ^2(x)+324 e^2 x^2 \log ^3(x)+\left (729 e^2 x^2-54 e^{2+x} x^4\right ) \log ^4(x)+\left (-1620 e^2 x+108 e^{2+x} x^3\right ) \log ^6(x)+\left (1215 e^2-81 e^{2+x} x^2\right ) \log ^8(x)}{576 x^4+240 x^5+25 x^6+e^{2 x} x^6+e^x \left (48 x^5+10 x^6\right )+\left (-4320 x^3-2340 x^4-300 x^5-12 e^{2 x} x^5+e^x \left (-468 x^4-120 x^5\right )\right ) \log ^2(x)+\left (14580 x^2+8910 x^3+1350 x^4+54 e^{2 x} x^4+e^x \left (1782 x^3+540 x^4\right )\right ) \log ^4(x)+\left (-24300 x-16200 x^2-2700 x^3-108 e^{2 x} x^3+e^x \left (-3240 x^2-1080 x^3\right )\right ) \log ^6(x)+\left (18225+12150 x+2025 x^2+81 e^{2 x} x^2+e^x \left (2430 x+810 x^2\right )\right ) \log ^8(x)} \, dx=\frac {9 \, x e^{4} \log \left (x\right )^{4} - 6 \, x^{2} e^{4} \log \left (x\right )^{2} + x^{3} e^{4}}{9 \, {\left (5 \, {\left (x + 3\right )} e^{2} + x e^{\left (x + 2\right )}\right )} \log \left (x\right )^{4} + x^{3} e^{\left (x + 2\right )} - 6 \, {\left (x^{2} e^{\left (x + 2\right )} + 5 \, {\left (x^{2} + 3 \, x\right )} e^{2}\right )} \log \left (x\right )^{2} + {\left (5 \, x^{3} + 24 \, x^{2}\right )} e^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (24) = 48\).
Time = 0.68 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.23 \[ \int \frac {24 e^2 x^4-e^{2+x} x^6-108 e^2 x^3 \log (x)+\left (-180 e^2 x^3+12 e^{2+x} x^5\right ) \log ^2(x)+324 e^2 x^2 \log ^3(x)+\left (729 e^2 x^2-54 e^{2+x} x^4\right ) \log ^4(x)+\left (-1620 e^2 x+108 e^{2+x} x^3\right ) \log ^6(x)+\left (1215 e^2-81 e^{2+x} x^2\right ) \log ^8(x)}{576 x^4+240 x^5+25 x^6+e^{2 x} x^6+e^x \left (48 x^5+10 x^6\right )+\left (-4320 x^3-2340 x^4-300 x^5-12 e^{2 x} x^5+e^x \left (-468 x^4-120 x^5\right )\right ) \log ^2(x)+\left (14580 x^2+8910 x^3+1350 x^4+54 e^{2 x} x^4+e^x \left (1782 x^3+540 x^4\right )\right ) \log ^4(x)+\left (-24300 x-16200 x^2-2700 x^3-108 e^{2 x} x^3+e^x \left (-3240 x^2-1080 x^3\right )\right ) \log ^6(x)+\left (18225+12150 x+2025 x^2+81 e^{2 x} x^2+e^x \left (2430 x+810 x^2\right )\right ) \log ^8(x)} \, dx=\frac {x^{3} e^{2} - 6 x^{2} e^{2} \log {\left (x \right )}^{2} + 9 x e^{2} \log {\left (x \right )}^{4}}{5 x^{3} - 30 x^{2} \log {\left (x \right )}^{2} + 24 x^{2} + 45 x \log {\left (x \right )}^{4} - 90 x \log {\left (x \right )}^{2} + \left (x^{3} - 6 x^{2} \log {\left (x \right )}^{2} + 9 x \log {\left (x \right )}^{4}\right ) e^{x} + 135 \log {\left (x \right )}^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (29) = 58\).
Time = 0.43 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.77 \[ \int \frac {24 e^2 x^4-e^{2+x} x^6-108 e^2 x^3 \log (x)+\left (-180 e^2 x^3+12 e^{2+x} x^5\right ) \log ^2(x)+324 e^2 x^2 \log ^3(x)+\left (729 e^2 x^2-54 e^{2+x} x^4\right ) \log ^4(x)+\left (-1620 e^2 x+108 e^{2+x} x^3\right ) \log ^6(x)+\left (1215 e^2-81 e^{2+x} x^2\right ) \log ^8(x)}{576 x^4+240 x^5+25 x^6+e^{2 x} x^6+e^x \left (48 x^5+10 x^6\right )+\left (-4320 x^3-2340 x^4-300 x^5-12 e^{2 x} x^5+e^x \left (-468 x^4-120 x^5\right )\right ) \log ^2(x)+\left (14580 x^2+8910 x^3+1350 x^4+54 e^{2 x} x^4+e^x \left (1782 x^3+540 x^4\right )\right ) \log ^4(x)+\left (-24300 x-16200 x^2-2700 x^3-108 e^{2 x} x^3+e^x \left (-3240 x^2-1080 x^3\right )\right ) \log ^6(x)+\left (18225+12150 x+2025 x^2+81 e^{2 x} x^2+e^x \left (2430 x+810 x^2\right )\right ) \log ^8(x)} \, dx=\frac {9 \, x e^{2} \log \left (x\right )^{4} - 6 \, x^{2} e^{2} \log \left (x\right )^{2} + x^{3} e^{2}}{45 \, {\left (x + 3\right )} \log \left (x\right )^{4} + 5 \, x^{3} - 30 \, {\left (x^{2} + 3 \, x\right )} \log \left (x\right )^{2} + 24 \, x^{2} + {\left (9 \, x \log \left (x\right )^{4} - 6 \, x^{2} \log \left (x\right )^{2} + x^{3}\right )} e^{x}} \]
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Timed out. \[ \int \frac {24 e^2 x^4-e^{2+x} x^6-108 e^2 x^3 \log (x)+\left (-180 e^2 x^3+12 e^{2+x} x^5\right ) \log ^2(x)+324 e^2 x^2 \log ^3(x)+\left (729 e^2 x^2-54 e^{2+x} x^4\right ) \log ^4(x)+\left (-1620 e^2 x+108 e^{2+x} x^3\right ) \log ^6(x)+\left (1215 e^2-81 e^{2+x} x^2\right ) \log ^8(x)}{576 x^4+240 x^5+25 x^6+e^{2 x} x^6+e^x \left (48 x^5+10 x^6\right )+\left (-4320 x^3-2340 x^4-300 x^5-12 e^{2 x} x^5+e^x \left (-468 x^4-120 x^5\right )\right ) \log ^2(x)+\left (14580 x^2+8910 x^3+1350 x^4+54 e^{2 x} x^4+e^x \left (1782 x^3+540 x^4\right )\right ) \log ^4(x)+\left (-24300 x-16200 x^2-2700 x^3-108 e^{2 x} x^3+e^x \left (-3240 x^2-1080 x^3\right )\right ) \log ^6(x)+\left (18225+12150 x+2025 x^2+81 e^{2 x} x^2+e^x \left (2430 x+810 x^2\right )\right ) \log ^8(x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {24 e^2 x^4-e^{2+x} x^6-108 e^2 x^3 \log (x)+\left (-180 e^2 x^3+12 e^{2+x} x^5\right ) \log ^2(x)+324 e^2 x^2 \log ^3(x)+\left (729 e^2 x^2-54 e^{2+x} x^4\right ) \log ^4(x)+\left (-1620 e^2 x+108 e^{2+x} x^3\right ) \log ^6(x)+\left (1215 e^2-81 e^{2+x} x^2\right ) \log ^8(x)}{576 x^4+240 x^5+25 x^6+e^{2 x} x^6+e^x \left (48 x^5+10 x^6\right )+\left (-4320 x^3-2340 x^4-300 x^5-12 e^{2 x} x^5+e^x \left (-468 x^4-120 x^5\right )\right ) \log ^2(x)+\left (14580 x^2+8910 x^3+1350 x^4+54 e^{2 x} x^4+e^x \left (1782 x^3+540 x^4\right )\right ) \log ^4(x)+\left (-24300 x-16200 x^2-2700 x^3-108 e^{2 x} x^3+e^x \left (-3240 x^2-1080 x^3\right )\right ) \log ^6(x)+\left (18225+12150 x+2025 x^2+81 e^{2 x} x^2+e^x \left (2430 x+810 x^2\right )\right ) \log ^8(x)} \, dx=-\int -\frac {{\ln \left (x\right )}^2\,\left (12\,x^5\,{\mathrm {e}}^{x+2}-180\,x^3\,{\mathrm {e}}^2\right )-{\ln \left (x\right )}^4\,\left (54\,x^4\,{\mathrm {e}}^{x+2}-729\,x^2\,{\mathrm {e}}^2\right )+{\ln \left (x\right )}^8\,\left (1215\,{\mathrm {e}}^2-81\,x^2\,{\mathrm {e}}^{x+2}\right )-x^6\,{\mathrm {e}}^{x+2}+24\,x^4\,{\mathrm {e}}^2-{\ln \left (x\right )}^6\,\left (1620\,x\,{\mathrm {e}}^2-108\,x^3\,{\mathrm {e}}^{x+2}\right )-108\,x^3\,{\mathrm {e}}^2\,\ln \left (x\right )+324\,x^2\,{\mathrm {e}}^2\,{\ln \left (x\right )}^3}{{\mathrm {e}}^x\,\left (10\,x^6+48\,x^5\right )-{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^x\,\left (120\,x^5+468\,x^4\right )+12\,x^5\,{\mathrm {e}}^{2\,x}+4320\,x^3+2340\,x^4+300\,x^5\right )+{\ln \left (x\right )}^4\,\left ({\mathrm {e}}^x\,\left (540\,x^4+1782\,x^3\right )+54\,x^4\,{\mathrm {e}}^{2\,x}+14580\,x^2+8910\,x^3+1350\,x^4\right )+{\ln \left (x\right )}^8\,\left (12150\,x+81\,x^2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (810\,x^2+2430\,x\right )+2025\,x^2+18225\right )+x^6\,{\mathrm {e}}^{2\,x}-{\ln \left (x\right )}^6\,\left (24300\,x+{\mathrm {e}}^x\,\left (1080\,x^3+3240\,x^2\right )+108\,x^3\,{\mathrm {e}}^{2\,x}+16200\,x^2+2700\,x^3\right )+576\,x^4+240\,x^5+25\,x^6} \,d x \]
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