Integrand size = 238, antiderivative size = 32 \[ \int \frac {e^{\frac {6}{-6561-11502 x-4717 x^2+1580 x^3+970 x^4-174 x^5-60 x^6+16 x^7-x^8+\log \left (\frac {3}{x}\right )}} \left (6+69012 x+56604 x^2-28440 x^3-23280 x^4+5220 x^5+2160 x^6-672 x^7+48 x^8\right )}{43046721 x+150929244 x^2+194192478 x^3+87777108 x^4-26824571 x^5-34936372 x^6-1864564 x^7+5877004 x^8+602158 x^9-655100 x^{10}-26130 x^{11}+48760 x^{12}-3908 x^{13}-1572 x^{14}+376 x^{15}-32 x^{16}+x^{17}+\left (-13122 x-23004 x^2-9434 x^3+3160 x^4+1940 x^5-348 x^6-120 x^7+32 x^8-2 x^9\right ) \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=e^{\frac {6}{-\left (-x+(9+(4-x) x)^2\right )^2+\log \left (\frac {3}{x}\right )}} \]
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Time = 3.86 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {6820, 12, 6838} \[ \int \frac {e^{\frac {6}{-6561-11502 x-4717 x^2+1580 x^3+970 x^4-174 x^5-60 x^6+16 x^7-x^8+\log \left (\frac {3}{x}\right )}} \left (6+69012 x+56604 x^2-28440 x^3-23280 x^4+5220 x^5+2160 x^6-672 x^7+48 x^8\right )}{43046721 x+150929244 x^2+194192478 x^3+87777108 x^4-26824571 x^5-34936372 x^6-1864564 x^7+5877004 x^8+602158 x^9-655100 x^{10}-26130 x^{11}+48760 x^{12}-3908 x^{13}-1572 x^{14}+376 x^{15}-32 x^{16}+x^{17}+\left (-13122 x-23004 x^2-9434 x^3+3160 x^4+1940 x^5-348 x^6-120 x^7+32 x^8-2 x^9\right ) \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=\exp \left (-\frac {6}{\left (x^4-8 x^3-2 x^2+71 x+81\right )^2-\log \left (\frac {3}{x}\right )}\right ) \]
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Rule 12
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \frac {6 \exp \left (-\frac {6}{\left (81+71 x-2 x^2-8 x^3+x^4\right )^2-\log \left (\frac {3}{x}\right )}\right ) \left (1+11502 x+9434 x^2-4740 x^3-3880 x^4+870 x^5+360 x^6-112 x^7+8 x^8\right )}{x \left (\left (81+71 x-2 x^2-8 x^3+x^4\right )^2-\log \left (\frac {3}{x}\right )\right )^2} \, dx \\ & = 6 \int \frac {\exp \left (-\frac {6}{\left (81+71 x-2 x^2-8 x^3+x^4\right )^2-\log \left (\frac {3}{x}\right )}\right ) \left (1+11502 x+9434 x^2-4740 x^3-3880 x^4+870 x^5+360 x^6-112 x^7+8 x^8\right )}{x \left (\left (81+71 x-2 x^2-8 x^3+x^4\right )^2-\log \left (\frac {3}{x}\right )\right )^2} \, dx \\ & = \exp \left (-\frac {6}{\left (81+71 x-2 x^2-8 x^3+x^4\right )^2-\log \left (\frac {3}{x}\right )}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {e^{\frac {6}{-6561-11502 x-4717 x^2+1580 x^3+970 x^4-174 x^5-60 x^6+16 x^7-x^8+\log \left (\frac {3}{x}\right )}} \left (6+69012 x+56604 x^2-28440 x^3-23280 x^4+5220 x^5+2160 x^6-672 x^7+48 x^8\right )}{43046721 x+150929244 x^2+194192478 x^3+87777108 x^4-26824571 x^5-34936372 x^6-1864564 x^7+5877004 x^8+602158 x^9-655100 x^{10}-26130 x^{11}+48760 x^{12}-3908 x^{13}-1572 x^{14}+376 x^{15}-32 x^{16}+x^{17}+\left (-13122 x-23004 x^2-9434 x^3+3160 x^4+1940 x^5-348 x^6-120 x^7+32 x^8-2 x^9\right ) \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=e^{-\frac {6}{\left (81+71 x-2 x^2-8 x^3+x^4\right )^2-\log \left (\frac {3}{x}\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.62
\[{\mathrm e}^{\frac {6}{\ln \left (\frac {3}{x}\right )-x^{8}+16 x^{7}-60 x^{6}-174 x^{5}+970 x^{4}+1580 x^{3}-4717 x^{2}-11502 x -6561}}\]
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Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\frac {6}{-6561-11502 x-4717 x^2+1580 x^3+970 x^4-174 x^5-60 x^6+16 x^7-x^8+\log \left (\frac {3}{x}\right )}} \left (6+69012 x+56604 x^2-28440 x^3-23280 x^4+5220 x^5+2160 x^6-672 x^7+48 x^8\right )}{43046721 x+150929244 x^2+194192478 x^3+87777108 x^4-26824571 x^5-34936372 x^6-1864564 x^7+5877004 x^8+602158 x^9-655100 x^{10}-26130 x^{11}+48760 x^{12}-3908 x^{13}-1572 x^{14}+376 x^{15}-32 x^{16}+x^{17}+\left (-13122 x-23004 x^2-9434 x^3+3160 x^4+1940 x^5-348 x^6-120 x^7+32 x^8-2 x^9\right ) \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (\frac {3}{x}\right ) + 6561}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.97 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {6}{-6561-11502 x-4717 x^2+1580 x^3+970 x^4-174 x^5-60 x^6+16 x^7-x^8+\log \left (\frac {3}{x}\right )}} \left (6+69012 x+56604 x^2-28440 x^3-23280 x^4+5220 x^5+2160 x^6-672 x^7+48 x^8\right )}{43046721 x+150929244 x^2+194192478 x^3+87777108 x^4-26824571 x^5-34936372 x^6-1864564 x^7+5877004 x^8+602158 x^9-655100 x^{10}-26130 x^{11}+48760 x^{12}-3908 x^{13}-1572 x^{14}+376 x^{15}-32 x^{16}+x^{17}+\left (-13122 x-23004 x^2-9434 x^3+3160 x^4+1940 x^5-348 x^6-120 x^7+32 x^8-2 x^9\right ) \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=e^{\frac {6}{- x^{8} + 16 x^{7} - 60 x^{6} - 174 x^{5} + 970 x^{4} + 1580 x^{3} - 4717 x^{2} - 11502 x + \log {\left (\frac {3}{x} \right )} - 6561}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (29) = 58\).
Time = 1.65 (sec) , antiderivative size = 859, normalized size of antiderivative = 26.84 \[ \int \frac {e^{\frac {6}{-6561-11502 x-4717 x^2+1580 x^3+970 x^4-174 x^5-60 x^6+16 x^7-x^8+\log \left (\frac {3}{x}\right )}} \left (6+69012 x+56604 x^2-28440 x^3-23280 x^4+5220 x^5+2160 x^6-672 x^7+48 x^8\right )}{43046721 x+150929244 x^2+194192478 x^3+87777108 x^4-26824571 x^5-34936372 x^6-1864564 x^7+5877004 x^8+602158 x^9-655100 x^{10}-26130 x^{11}+48760 x^{12}-3908 x^{13}-1572 x^{14}+376 x^{15}-32 x^{16}+x^{17}+\left (-13122 x-23004 x^2-9434 x^3+3160 x^4+1940 x^5-348 x^6-120 x^7+32 x^8-2 x^9\right ) \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=\frac {8 \, x^{8} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (3\right ) + \log \left (x\right ) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} - \frac {112 \, x^{7} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (3\right ) + \log \left (x\right ) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} + \frac {360 \, x^{6} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (3\right ) + \log \left (x\right ) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} + \frac {870 \, x^{5} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (3\right ) + \log \left (x\right ) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} - \frac {3880 \, x^{4} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (3\right ) + \log \left (x\right ) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} - \frac {4740 \, x^{3} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (3\right ) + \log \left (x\right ) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} + \frac {9434 \, x^{2} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (3\right ) + \log \left (x\right ) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} + \frac {11502 \, x e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (3\right ) + \log \left (x\right ) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} + \frac {e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (3\right ) + \log \left (x\right ) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} \]
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Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\frac {6}{-6561-11502 x-4717 x^2+1580 x^3+970 x^4-174 x^5-60 x^6+16 x^7-x^8+\log \left (\frac {3}{x}\right )}} \left (6+69012 x+56604 x^2-28440 x^3-23280 x^4+5220 x^5+2160 x^6-672 x^7+48 x^8\right )}{43046721 x+150929244 x^2+194192478 x^3+87777108 x^4-26824571 x^5-34936372 x^6-1864564 x^7+5877004 x^8+602158 x^9-655100 x^{10}-26130 x^{11}+48760 x^{12}-3908 x^{13}-1572 x^{14}+376 x^{15}-32 x^{16}+x^{17}+\left (-13122 x-23004 x^2-9434 x^3+3160 x^4+1940 x^5-348 x^6-120 x^7+32 x^8-2 x^9\right ) \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (\frac {3}{x}\right ) + 6561}\right )} \]
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Time = 8.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\frac {6}{-6561-11502 x-4717 x^2+1580 x^3+970 x^4-174 x^5-60 x^6+16 x^7-x^8+\log \left (\frac {3}{x}\right )}} \left (6+69012 x+56604 x^2-28440 x^3-23280 x^4+5220 x^5+2160 x^6-672 x^7+48 x^8\right )}{43046721 x+150929244 x^2+194192478 x^3+87777108 x^4-26824571 x^5-34936372 x^6-1864564 x^7+5877004 x^8+602158 x^9-655100 x^{10}-26130 x^{11}+48760 x^{12}-3908 x^{13}-1572 x^{14}+376 x^{15}-32 x^{16}+x^{17}+\left (-13122 x-23004 x^2-9434 x^3+3160 x^4+1940 x^5-348 x^6-120 x^7+32 x^8-2 x^9\right ) \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx={\mathrm {e}}^{-\frac {6}{11502\,x-\ln \left (\frac {3}{x}\right )+4717\,x^2-1580\,x^3-970\,x^4+174\,x^5+60\,x^6-16\,x^7+x^8+6561}} \]
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