Integrand size = 147, antiderivative size = 28 \[ \int \frac {-54 e^6 x^3+\left (-864 e^2+e^4 \left (-1296 x^2-540 x^3\right )\right ) \log (x)+e^2 \left (864-10368 x-8640 x^2-1800 x^3\right ) \log ^2(x)+\left (-24768-34560 x-14400 x^2-2000 x^3\right ) \log ^3(x)}{27 e^6 x^3+e^4 \left (648 x^2+270 x^3\right ) \log (x)+e^2 \left (5184 x+4320 x^2+900 x^3\right ) \log ^2(x)+\left (13824+17280 x+7200 x^2+1000 x^3\right ) \log ^3(x)} \, dx=1-2 x-\frac {4}{\left (-4-\frac {5 x}{3}-\frac {e^2 x}{2 \log (x)}\right )^2} \]
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\[ \int \frac {-54 e^6 x^3+\left (-864 e^2+e^4 \left (-1296 x^2-540 x^3\right )\right ) \log (x)+e^2 \left (864-10368 x-8640 x^2-1800 x^3\right ) \log ^2(x)+\left (-24768-34560 x-14400 x^2-2000 x^3\right ) \log ^3(x)}{27 e^6 x^3+e^4 \left (648 x^2+270 x^3\right ) \log (x)+e^2 \left (5184 x+4320 x^2+900 x^3\right ) \log ^2(x)+\left (13824+17280 x+7200 x^2+1000 x^3\right ) \log ^3(x)} \, dx=\int \frac {-54 e^6 x^3+\left (-864 e^2+e^4 \left (-1296 x^2-540 x^3\right )\right ) \log (x)+e^2 \left (864-10368 x-8640 x^2-1800 x^3\right ) \log ^2(x)+\left (-24768-34560 x-14400 x^2-2000 x^3\right ) \log ^3(x)}{27 e^6 x^3+e^4 \left (648 x^2+270 x^3\right ) \log (x)+e^2 \left (5184 x+4320 x^2+900 x^3\right ) \log ^2(x)+\left (13824+17280 x+7200 x^2+1000 x^3\right ) \log ^3(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (-27 e^6 x^3-54 e^2 \left (8+e^2 x^2 (12+5 x)\right ) \log (x)-36 e^2 \left (-12+144 x+120 x^2+25 x^3\right ) \log ^2(x)-8 \left (1548+2160 x+900 x^2+125 x^3\right ) \log ^3(x)\right )}{\left (3 e^2 x+2 (12+5 x) \log (x)\right )^3} \, dx \\ & = 2 \int \frac {-27 e^6 x^3-54 e^2 \left (8+e^2 x^2 (12+5 x)\right ) \log (x)-36 e^2 \left (-12+144 x+120 x^2+25 x^3\right ) \log ^2(x)-8 \left (1548+2160 x+900 x^2+125 x^3\right ) \log ^3(x)}{\left (3 e^2 x+2 (12+5 x) \log (x)\right )^3} \, dx \\ & = 2 \int \left (\frac {-1548-2160 x-900 x^2-125 x^3}{(12+5 x)^3}+\frac {648 e^4 x \left (144+6 \left (20+3 e^2\right ) x+25 x^2\right )}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3}+\frac {108 e^2 \left (-288-24 \left (10+3 e^2\right ) x-5 \left (10-3 e^2\right ) x^2\right )}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2}-\frac {216 e^2 (-6+5 x)}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )}\right ) \, dx \\ & = 2 \int \frac {-1548-2160 x-900 x^2-125 x^3}{(12+5 x)^3} \, dx+\left (216 e^2\right ) \int \frac {-288-24 \left (10+3 e^2\right ) x-5 \left (10-3 e^2\right ) x^2}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2} \, dx-\left (432 e^2\right ) \int \frac {-6+5 x}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )} \, dx+\left (1296 e^4\right ) \int \frac {x \left (144+6 \left (20+3 e^2\right ) x+25 x^2\right )}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3} \, dx \\ & = 2 \int \left (-1+\frac {180}{(12+5 x)^3}\right ) \, dx+\left (216 e^2\right ) \int \left (\frac {1296 e^2}{5 (12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2}-\frac {144 e^2}{5 (12+5 x)^2 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2}+\frac {-10+3 e^2}{5 (12+5 x) \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2}\right ) \, dx-\left (432 e^2\right ) \int \left (-\frac {18}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )}+\frac {1}{(12+5 x)^2 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )}\right ) \, dx+\left (1296 e^4\right ) \int \left (\frac {1}{5 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3}+\frac {2592 e^2}{25 (12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3}-\frac {432 e^2}{25 (12+5 x)^2 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3}+\frac {6 \left (-10+3 e^2\right )}{25 (12+5 x) \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3}\right ) \, dx \\ & = -2 x-\frac {36}{(12+5 x)^2}-\left (432 e^2\right ) \int \frac {1}{(12+5 x)^2 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )} \, dx+\left (7776 e^2\right ) \int \frac {1}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )} \, dx+\frac {1}{5} \left (1296 e^4\right ) \int \frac {1}{\left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3} \, dx-\frac {1}{5} \left (31104 e^4\right ) \int \frac {1}{(12+5 x)^2 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2} \, dx+\frac {1}{5} \left (279936 e^4\right ) \int \frac {1}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2} \, dx-\frac {1}{25} \left (559872 e^6\right ) \int \frac {1}{(12+5 x)^2 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3} \, dx+\frac {1}{25} \left (3359232 e^6\right ) \int \frac {1}{(12+5 x)^3 \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3} \, dx-\frac {1}{5} \left (216 e^2 \left (10-3 e^2\right )\right ) \int \frac {1}{(12+5 x) \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^2} \, dx-\frac {1}{25} \left (7776 e^4 \left (10-3 e^2\right )\right ) \int \frac {1}{(12+5 x) \left (3 e^2 x+24 \log (x)+10 x \log (x)\right )^3} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(28)=56\).
Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {-54 e^6 x^3+\left (-864 e^2+e^4 \left (-1296 x^2-540 x^3\right )\right ) \log (x)+e^2 \left (864-10368 x-8640 x^2-1800 x^3\right ) \log ^2(x)+\left (-24768-34560 x-14400 x^2-2000 x^3\right ) \log ^3(x)}{27 e^6 x^3+e^4 \left (648 x^2+270 x^3\right ) \log (x)+e^2 \left (5184 x+4320 x^2+900 x^3\right ) \log ^2(x)+\left (13824+17280 x+7200 x^2+1000 x^3\right ) \log ^3(x)} \, dx=-\frac {2 \left (9 e^4 x^3+12 e^2 x^2 (12+5 x) \log (x)+4 \left (18+144 x+120 x^2+25 x^3\right ) \log ^2(x)\right )}{\left (3 e^2 x+2 (12+5 x) \log (x)\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(23)=46\).
Time = 1.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86
method | result | size |
risch | \(-\frac {2 \left (25 x^{3}+120 x^{2}+144 x +18\right )}{25 x^{2}+120 x +144}+\frac {108 \left (3 \,{\mathrm e}^{2} x +20 x \ln \left (x \right )+48 \ln \left (x \right )\right ) {\mathrm e}^{2} x}{\left (25 x^{2}+120 x +144\right ) \left (3 \,{\mathrm e}^{2} x +10 x \ln \left (x \right )+24 \ln \left (x \right )\right )^{2}}\) | \(80\) |
norman | \(\frac {\frac {26928 \ln \left (x \right )^{2}}{5}+\frac {432 x^{2} {\mathrm e}^{4}}{5}+3456 x \ln \left (x \right )^{2}+288 x^{2} {\mathrm e}^{2} \ln \left (x \right )+\frac {6912 x \,{\mathrm e}^{2} \ln \left (x \right )}{5}-18 x^{3} {\mathrm e}^{4}-200 x^{3} \ln \left (x \right )^{2}-120 x^{3} {\mathrm e}^{2} \ln \left (x \right )}{\left (3 \,{\mathrm e}^{2} x +10 x \ln \left (x \right )+24 \ln \left (x \right )\right )^{2}}\) | \(85\) |
default | \(-\frac {2 \left (-\frac {13464 \ln \left (x \right )^{2}}{5}-\frac {216 x^{2} {\mathrm e}^{4}}{5}-1728 x \ln \left (x \right )^{2}-\frac {3456 x \,{\mathrm e}^{2} \ln \left (x \right )}{5}-144 x^{2} {\mathrm e}^{2} \ln \left (x \right )+100 x^{3} \ln \left (x \right )^{2}+9 x^{3} {\mathrm e}^{4}+60 x^{3} {\mathrm e}^{2} \ln \left (x \right )\right )}{\left (3 \,{\mathrm e}^{2} x +10 x \ln \left (x \right )+24 \ln \left (x \right )\right )^{2}}\) | \(86\) |
parallelrisch | \(\frac {-12000 x^{3} {\mathrm e}^{2} \ln \left (x \right )-1800 x^{3} {\mathrm e}^{4}+8640 x^{2} {\mathrm e}^{4}+345600 x \ln \left (x \right )^{2}-20000 x^{3} \ln \left (x \right )^{2}+138240 x \,{\mathrm e}^{2} \ln \left (x \right )+28800 x^{2} {\mathrm e}^{2} \ln \left (x \right )+538560 \ln \left (x \right )^{2}}{900 x^{2} {\mathrm e}^{4}+6000 x^{2} {\mathrm e}^{2} \ln \left (x \right )+10000 x^{2} \ln \left (x \right )^{2}+14400 x \,{\mathrm e}^{2} \ln \left (x \right )+48000 x \ln \left (x \right )^{2}+57600 \ln \left (x \right )^{2}}\) | \(119\) |
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Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.18 \[ \int \frac {-54 e^6 x^3+\left (-864 e^2+e^4 \left (-1296 x^2-540 x^3\right )\right ) \log (x)+e^2 \left (864-10368 x-8640 x^2-1800 x^3\right ) \log ^2(x)+\left (-24768-34560 x-14400 x^2-2000 x^3\right ) \log ^3(x)}{27 e^6 x^3+e^4 \left (648 x^2+270 x^3\right ) \log (x)+e^2 \left (5184 x+4320 x^2+900 x^3\right ) \log ^2(x)+\left (13824+17280 x+7200 x^2+1000 x^3\right ) \log ^3(x)} \, dx=-\frac {2 \, {\left (9 \, x^{3} e^{4} + 12 \, {\left (5 \, x^{3} + 12 \, x^{2}\right )} e^{2} \log \left (x\right ) + 4 \, {\left (25 \, x^{3} + 120 \, x^{2} + 144 \, x + 18\right )} \log \left (x\right )^{2}\right )}}{9 \, x^{2} e^{4} + 12 \, {\left (5 \, x^{2} + 12 \, x\right )} e^{2} \log \left (x\right ) + 4 \, {\left (25 \, x^{2} + 120 \, x + 144\right )} \log \left (x\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.57 \[ \int \frac {-54 e^6 x^3+\left (-864 e^2+e^4 \left (-1296 x^2-540 x^3\right )\right ) \log (x)+e^2 \left (864-10368 x-8640 x^2-1800 x^3\right ) \log ^2(x)+\left (-24768-34560 x-14400 x^2-2000 x^3\right ) \log ^3(x)}{27 e^6 x^3+e^4 \left (648 x^2+270 x^3\right ) \log (x)+e^2 \left (5184 x+4320 x^2+900 x^3\right ) \log ^2(x)+\left (13824+17280 x+7200 x^2+1000 x^3\right ) \log ^3(x)} \, dx=- 2 x + \frac {324 x^{2} e^{4} + \left (2160 x^{2} e^{2} + 5184 x e^{2}\right ) \log {\left (x \right )}}{225 x^{4} e^{4} + 1080 x^{3} e^{4} + 1296 x^{2} e^{4} + \left (1500 x^{4} e^{2} + 10800 x^{3} e^{2} + 25920 x^{2} e^{2} + 20736 x e^{2}\right ) \log {\left (x \right )} + \left (2500 x^{4} + 24000 x^{3} + 86400 x^{2} + 138240 x + 82944\right ) \log {\left (x \right )}^{2}} - \frac {36}{25 x^{2} + 120 x + 144} \]
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Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.32 \[ \int \frac {-54 e^6 x^3+\left (-864 e^2+e^4 \left (-1296 x^2-540 x^3\right )\right ) \log (x)+e^2 \left (864-10368 x-8640 x^2-1800 x^3\right ) \log ^2(x)+\left (-24768-34560 x-14400 x^2-2000 x^3\right ) \log ^3(x)}{27 e^6 x^3+e^4 \left (648 x^2+270 x^3\right ) \log (x)+e^2 \left (5184 x+4320 x^2+900 x^3\right ) \log ^2(x)+\left (13824+17280 x+7200 x^2+1000 x^3\right ) \log ^3(x)} \, dx=-\frac {2 \, {\left (9 \, x^{3} e^{4} + 4 \, {\left (25 \, x^{3} + 120 \, x^{2} + 144 \, x + 18\right )} \log \left (x\right )^{2} + 12 \, {\left (5 \, x^{3} e^{2} + 12 \, x^{2} e^{2}\right )} \log \left (x\right )\right )}}{9 \, x^{2} e^{4} + 4 \, {\left (25 \, x^{2} + 120 \, x + 144\right )} \log \left (x\right )^{2} + 12 \, {\left (5 \, x^{2} e^{2} + 12 \, x e^{2}\right )} \log \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (23) = 46\).
Time = 0.40 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.82 \[ \int \frac {-54 e^6 x^3+\left (-864 e^2+e^4 \left (-1296 x^2-540 x^3\right )\right ) \log (x)+e^2 \left (864-10368 x-8640 x^2-1800 x^3\right ) \log ^2(x)+\left (-24768-34560 x-14400 x^2-2000 x^3\right ) \log ^3(x)}{27 e^6 x^3+e^4 \left (648 x^2+270 x^3\right ) \log (x)+e^2 \left (5184 x+4320 x^2+900 x^3\right ) \log ^2(x)+\left (13824+17280 x+7200 x^2+1000 x^3\right ) \log ^3(x)} \, dx=-\frac {2 \, {\left (60 \, x^{3} e^{2} \log \left (x\right ) + 100 \, x^{3} \log \left (x\right )^{2} + 9 \, x^{3} e^{4} + 144 \, x^{2} e^{2} \log \left (x\right ) + 480 \, x^{2} \log \left (x\right )^{2} + 576 \, x \log \left (x\right )^{2} + 72 \, \log \left (x\right )^{2}\right )}}{60 \, x^{2} e^{2} \log \left (x\right ) + 100 \, x^{2} \log \left (x\right )^{2} + 9 \, x^{2} e^{4} + 144 \, x e^{2} \log \left (x\right ) + 480 \, x \log \left (x\right )^{2} + 576 \, \log \left (x\right )^{2}} \]
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Timed out. \[ \int \frac {-54 e^6 x^3+\left (-864 e^2+e^4 \left (-1296 x^2-540 x^3\right )\right ) \log (x)+e^2 \left (864-10368 x-8640 x^2-1800 x^3\right ) \log ^2(x)+\left (-24768-34560 x-14400 x^2-2000 x^3\right ) \log ^3(x)}{27 e^6 x^3+e^4 \left (648 x^2+270 x^3\right ) \log (x)+e^2 \left (5184 x+4320 x^2+900 x^3\right ) \log ^2(x)+\left (13824+17280 x+7200 x^2+1000 x^3\right ) \log ^3(x)} \, dx=\int -\frac {{\ln \left (x\right )}^3\,\left (2000\,x^3+14400\,x^2+34560\,x+24768\right )+54\,x^3\,{\mathrm {e}}^6+\ln \left (x\right )\,\left (864\,{\mathrm {e}}^2+{\mathrm {e}}^4\,\left (540\,x^3+1296\,x^2\right )\right )+{\mathrm {e}}^2\,{\ln \left (x\right )}^2\,\left (1800\,x^3+8640\,x^2+10368\,x-864\right )}{{\ln \left (x\right )}^3\,\left (1000\,x^3+7200\,x^2+17280\,x+13824\right )+27\,x^3\,{\mathrm {e}}^6+{\mathrm {e}}^4\,\ln \left (x\right )\,\left (270\,x^3+648\,x^2\right )+{\mathrm {e}}^2\,{\ln \left (x\right )}^2\,\left (900\,x^3+4320\,x^2+5184\,x\right )} \,d x \]
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