\(\int \frac {e^{\frac {28561-3042 e x+81 e^2 x^2+e^{30/x} (2426112-11664 e x-8748 e^2 x^2)+e^{10/x} (316368+1404 e x-972 e^2 x^2)+e^{20/x} (1314144+50382 e x+4374 e^2 x^2)+e^{40/x} (1679616-209952 e x+6561 e^2 x^2)}{81-972 e^{10/x}+4374 e^{20/x}-8748 e^{30/x}+6561 e^{40/x}}} (1014 e x^2-54 e^2 x^3+e^{50/x} (-209952 e x^2+13122 e^2 x^3)+e^{20/x} (18252000-4860 e^2 x^3+e (378000 x-15390 x^2))+e^{10/x} (2197000+810 e^2 x^3+e (-117000 x-3510 x^2))+e^{30/x} (50544000+14580 e^2 x^3+e (891000 x+54270 x^2))+e^{40/x} (46656000-21870 e^2 x^3+e (-2916000 x+58320 x^2)))}{-27 x^2+405 e^{10/x} x^2-2430 e^{20/x} x^2+7290 e^{30/x} x^2-10935 e^{40/x} x^2+6561 e^{50/x} x^2} \, dx\) [253]

Optimal result

Integrand size = 368, antiderivative size = 30 \[ \int \frac {e^{\frac {28561-3042 e x+81 e^2 x^2+e^{30/x} \left (2426112-11664 e x-8748 e^2 x^2\right )+e^{10/x} \left (316368+1404 e x-972 e^2 x^2\right )+e^{20/x} \left (1314144+50382 e x+4374 e^2 x^2\right )+e^{40/x} \left (1679616-209952 e x+6561 e^2 x^2\right )}{81-972 e^{10/x}+4374 e^{20/x}-8748 e^{30/x}+6561 e^{40/x}}} \left (1014 e x^2-54 e^2 x^3+e^{50/x} \left (-209952 e x^2+13122 e^2 x^3\right )+e^{20/x} \left (18252000-4860 e^2 x^3+e \left (378000 x-15390 x^2\right )\right )+e^{10/x} \left (2197000+810 e^2 x^3+e \left (-117000 x-3510 x^2\right )\right )+e^{30/x} \left (50544000+14580 e^2 x^3+e \left (891000 x+54270 x^2\right )\right )+e^{40/x} \left (46656000-21870 e^2 x^3+e \left (-2916000 x+58320 x^2\right )\right )\right )}{-27 x^2+405 e^{10/x} x^2-2430 e^{20/x} x^2+7290 e^{30/x} x^2-10935 e^{40/x} x^2+6561 e^{50/x} x^2} \, dx=e^{\left (\left (4-\frac {25}{3 \left (1-3 e^{10/x}\right )}\right )^2-e x\right )^2} \] Output:

exp(((4-5/(3/5-9/5*exp(5/x)^2))^2-x*exp(1))^2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(30)=60\).

Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.30 \[ \int \frac {e^{\frac {28561-3042 e x+81 e^2 x^2+e^{30/x} \left (2426112-11664 e x-8748 e^2 x^2\right )+e^{10/x} \left (316368+1404 e x-972 e^2 x^2\right )+e^{20/x} \left (1314144+50382 e x+4374 e^2 x^2\right )+e^{40/x} \left (1679616-209952 e x+6561 e^2 x^2\right )}{81-972 e^{10/x}+4374 e^{20/x}-8748 e^{30/x}+6561 e^{40/x}}} \left (1014 e x^2-54 e^2 x^3+e^{50/x} \left (-209952 e x^2+13122 e^2 x^3\right )+e^{20/x} \left (18252000-4860 e^2 x^3+e \left (378000 x-15390 x^2\right )\right )+e^{10/x} \left (2197000+810 e^2 x^3+e \left (-117000 x-3510 x^2\right )\right )+e^{30/x} \left (50544000+14580 e^2 x^3+e \left (891000 x+54270 x^2\right )\right )+e^{40/x} \left (46656000-21870 e^2 x^3+e \left (-2916000 x+58320 x^2\right )\right )\right )}{-27 x^2+405 e^{10/x} x^2-2430 e^{20/x} x^2+7290 e^{30/x} x^2-10935 e^{40/x} x^2+6561 e^{50/x} x^2} \, dx=e^{\frac {\left (169+936 e^{10/x}+1296 e^{20/x}-9 e x+54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x\right )^2}{81 \left (1-3 e^{10/x}\right )^4}} \] Input:

Integrate[(E^((28561 - 3042*E*x + 81*E^2*x^2 + E^(30/x)*(2426112 - 11664*E 
*x - 8748*E^2*x^2) + E^(10/x)*(316368 + 1404*E*x - 972*E^2*x^2) + E^(20/x) 
*(1314144 + 50382*E*x + 4374*E^2*x^2) + E^(40/x)*(1679616 - 209952*E*x + 6 
561*E^2*x^2))/(81 - 972*E^(10/x) + 4374*E^(20/x) - 8748*E^(30/x) + 6561*E^ 
(40/x)))*(1014*E*x^2 - 54*E^2*x^3 + E^(50/x)*(-209952*E*x^2 + 13122*E^2*x^ 
3) + E^(20/x)*(18252000 - 4860*E^2*x^3 + E*(378000*x - 15390*x^2)) + E^(10 
/x)*(2197000 + 810*E^2*x^3 + E*(-117000*x - 3510*x^2)) + E^(30/x)*(5054400 
0 + 14580*E^2*x^3 + E*(891000*x + 54270*x^2)) + E^(40/x)*(46656000 - 21870 
*E^2*x^3 + E*(-2916000*x + 58320*x^2))))/(-27*x^2 + 405*E^(10/x)*x^2 - 243 
0*E^(20/x)*x^2 + 7290*E^(30/x)*x^2 - 10935*E^(40/x)*x^2 + 6561*E^(50/x)*x^ 
2),x]
 

Output:

E^((169 + 936*E^(10/x) + 1296*E^(20/x) - 9*E*x + 54*E^(1 + 10/x)*x - 81*E^ 
(1 + 20/x)*x)^2/(81*(1 - 3*E^(10/x))^4))
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (-54 e^2 x^3+1014 e x^2+e^{50/x} \left (13122 e^2 x^3-209952 e x^2\right )+e^{20/x} \left (-4860 e^2 x^3+e \left (378000 x-15390 x^2\right )[phone]\right )+e^{10/x} \left (810 e^2 x^3+e \left (-3510 x^2-117000 x\right )+2197000\right )+e^{30/x} \left (14580 e^2 x^3+e \left (54270 x^2+891000 x\right )[phone]\right )+e^{40/x} \left (-21870 e^2 x^3+e \left (58320 x^2-2916000 x\right )[phone]\right )\right ) \exp \left (\frac {81 e^2 x^2+e^{30/x} \left (-8748 e^2 x^2-11664 e x+2426112\right )+e^{10/x} \left (-972 e^2 x^2+1404 e x+316368\right )+e^{20/x} \left (4374 e^2 x^2+50382 e x+1314144\right )+e^{40/x} \left (6561 e^2 x^2-209952 e x+1679616\right )-3042 e x+28561}{-972 e^{10/x}+4374 e^{20/x}-8748 e^{30/x}+6561 e^{40/x}+81}\right )}{405 e^{10/x} x^2-2430 e^{20/x} x^2+7290 e^{30/x} x^2-10935 e^{40/x} x^2+6561 e^{50/x} x^2-27 x^2} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (54 e^2 x^3-1014 e x^2-e^{50/x} \left (13122 e^2 x^3-209952 e x^2\right )-e^{20/x} \left (-4860 e^2 x^3+e \left (378000 x-15390 x^2\right )[phone]\right )-e^{10/x} \left (810 e^2 x^3+e \left (-3510 x^2-117000 x\right )+2197000\right )-e^{30/x} \left (14580 e^2 x^3+e \left (54270 x^2+891000 x\right )[phone]\right )-e^{40/x} \left (-21870 e^2 x^3+e \left (58320 x^2-2916000 x\right )[phone]\right )\right ) \exp \left (\frac {\left (-54 e^{\frac {10}{x}+1} x+81 e^{\frac {20}{x}+1} x+9 e x-936 e^{10/x}-1296 e^{20/x}-169\right )^2}{81 \left (3 e^{10/x}-1\right )^4}\right )}{27 \left (1-3 e^{10/x}\right )^5 x^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{27} \int -\frac {2 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (-27 e^2 x^3+507 e x^2-6561 e^{50/x} \left (16 e x^2-e^2 x^3\right )+135 e^{20/x} \left (-18 e^2 x^3+e \left (1400 x-57 x^2\right )+67600\right )+3645 e^{40/x} \left (-3 e^2 x^3-8 e \left (50 x-x^2\right )+6400\right )+5 e^{10/x} \left (81 e^2 x^3-117 e \left (3 x^2+100 x\right )+219700\right )+405 e^{30/x} \left (18 e^2 x^3+e \left (67 x^2+1100 x\right )+62400\right )\right )}{\left (1-3 e^{10/x}\right )^5 x^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{27} \int \frac {\exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (-27 e^2 x^3+507 e x^2-6561 e^{50/x} \left (16 e x^2-e^2 x^3\right )+135 e^{20/x} \left (-18 e^2 x^3+e \left (1400 x-57 x^2\right )+67600\right )+3645 e^{40/x} \left (-3 e^2 x^3-8 e \left (50 x-x^2\right )+6400\right )+5 e^{10/x} \left (81 e^2 x^3-117 e \left (3 x^2+100 x\right )+219700\right )+405 e^{30/x} \left (18 e^2 x^3+e \left (67 x^2+1100 x\right )+62400\right )\right )}{\left (1-3 e^{10/x}\right )^5 x^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2}{27} \int \left (\frac {37500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) (e x-148)}{\left (-1+3 e^{10/x}\right )^3 x^2}-27 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+1\right ) (e x-16)+\frac {1800 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (e x^2+10 e x-160\right )}{\left (-1+3 e^{10/x}\right ) x^2}+\frac {375 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (5 e x^2+148 e x-5568\right )}{\left (-1+3 e^{10/x}\right )^2 x^2}-\frac {19062500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right )}{3 \left (-1+3 e^{10/x}\right )^4 x^2}-\frac {7812500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right )}{3 \left (-1+3 e^{10/x}\right )^5 x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {2}{27} \int \frac {1}{3} \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (-\frac {1500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5 x^2}+\frac {9 e \left (108 e^{20/x} x-13 x+3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3 x}-81 e^2 x\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{81} \int 3 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (1-3 e^{10/x}\right )^5 x^2}+\frac {3 e \left (-108 e^{20/x} x+13 x-3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (1-3 e^{10/x}\right )^3 x}-27 e^2 x\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{27} \int \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (1-3 e^{10/x}\right )^5 x^2}+\frac {3 e \left (-108 e^{20/x} x+13 x-3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (1-3 e^{10/x}\right )^3 x}-27 e^2 x\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2}{27} \int \left (-\frac {500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+\frac {10}{x}\right ) \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5 x^2}+\frac {3 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+1\right ) \left (3 e^{10/x} x+108 e^{20/x} x-13 x+1500 e^{10/x}\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3 x}-27 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+2\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {2}{27} \int \frac {\exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (-\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5}+\frac {3 e x \left (108 e^{20/x} x-13 x+3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3}-27 e^2 x^3\right )}{x^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2}{27} \int \left (\frac {37500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) (e x-148)}{\left (-1+3 e^{10/x}\right )^3 x^2}-27 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+1\right ) (e x-16)+\frac {1800 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (e x^2+10 e x-160\right )}{\left (-1+3 e^{10/x}\right ) x^2}+\frac {375 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (5 e x^2+148 e x-5568\right )}{\left (-1+3 e^{10/x}\right )^2 x^2}-\frac {19062500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right )}{3 \left (-1+3 e^{10/x}\right )^4 x^2}-\frac {7812500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right )}{3 \left (-1+3 e^{10/x}\right )^5 x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {2}{27} \int \frac {1}{3} \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (-\frac {1500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5 x^2}+\frac {9 e \left (108 e^{20/x} x-13 x+3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3 x}-81 e^2 x\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{81} \int 3 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (1-3 e^{10/x}\right )^5 x^2}+\frac {3 e \left (-108 e^{20/x} x+13 x-3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (1-3 e^{10/x}\right )^3 x}-27 e^2 x\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{27} \int \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (1-3 e^{10/x}\right )^5 x^2}+\frac {3 e \left (-108 e^{20/x} x+13 x-3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (1-3 e^{10/x}\right )^3 x}-27 e^2 x\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2}{27} \int \left (-\frac {500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+\frac {10}{x}\right ) \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5 x^2}+\frac {3 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+1\right ) \left (3 e^{10/x} x+108 e^{20/x} x-13 x+1500 e^{10/x}\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3 x}-27 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+2\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {2}{27} \int \frac {\exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (-\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5}+\frac {3 e x \left (108 e^{20/x} x-13 x+3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3}-27 e^2 x^3\right )}{x^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2}{27} \int \left (\frac {37500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) (e x-148)}{\left (-1+3 e^{10/x}\right )^3 x^2}-27 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+1\right ) (e x-16)+\frac {1800 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (e x^2+10 e x-160\right )}{\left (-1+3 e^{10/x}\right ) x^2}+\frac {375 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (5 e x^2+148 e x-5568\right )}{\left (-1+3 e^{10/x}\right )^2 x^2}-\frac {19062500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right )}{3 \left (-1+3 e^{10/x}\right )^4 x^2}-\frac {7812500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right )}{3 \left (-1+3 e^{10/x}\right )^5 x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {2}{27} \int \frac {1}{3} \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (-\frac {1500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5 x^2}+\frac {9 e \left (108 e^{20/x} x-13 x+3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3 x}-81 e^2 x\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{81} \int 3 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (1-3 e^{10/x}\right )^5 x^2}+\frac {3 e \left (-108 e^{20/x} x+13 x-3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (1-3 e^{10/x}\right )^3 x}-27 e^2 x\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{27} \int \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (1-3 e^{10/x}\right )^5 x^2}+\frac {3 e \left (-108 e^{20/x} x+13 x-3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (1-3 e^{10/x}\right )^3 x}-27 e^2 x\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2}{27} \int \left (-\frac {500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+\frac {10}{x}\right ) \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5 x^2}+\frac {3 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+1\right ) \left (3 e^{10/x} x+108 e^{20/x} x-13 x+1500 e^{10/x}\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3 x}-27 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+2\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {2}{27} \int \frac {\exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (-\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5}+\frac {3 e x \left (108 e^{20/x} x-13 x+3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3}-27 e^2 x^3\right )}{x^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2}{27} \int \left (\frac {37500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) (e x-148)}{\left (-1+3 e^{10/x}\right )^3 x^2}-27 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+1\right ) (e x-16)+\frac {1800 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (e x^2+10 e x-160\right )}{\left (-1+3 e^{10/x}\right ) x^2}+\frac {375 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (5 e x^2+148 e x-5568\right )}{\left (-1+3 e^{10/x}\right )^2 x^2}-\frac {19062500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right )}{3 \left (-1+3 e^{10/x}\right )^4 x^2}-\frac {7812500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right )}{3 \left (-1+3 e^{10/x}\right )^5 x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {2}{27} \int \frac {1}{3} \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (-\frac {1500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5 x^2}+\frac {9 e \left (108 e^{20/x} x-13 x+3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3 x}-81 e^2 x\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{81} \int 3 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (1-3 e^{10/x}\right )^5 x^2}+\frac {3 e \left (-108 e^{20/x} x+13 x-3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (1-3 e^{10/x}\right )^3 x}-27 e^2 x\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{27} \int \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (1-3 e^{10/x}\right )^5 x^2}+\frac {3 e \left (-108 e^{20/x} x+13 x-3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (1-3 e^{10/x}\right )^3 x}-27 e^2 x\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2}{27} \int \left (-\frac {500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+\frac {10}{x}\right ) \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5 x^2}+\frac {3 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+1\right ) \left (3 e^{10/x} x+108 e^{20/x} x-13 x+1500 e^{10/x}\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3 x}-27 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+2\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {2}{27} \int \frac {\exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (-\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5}+\frac {3 e x \left (108 e^{20/x} x-13 x+3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3}-27 e^2 x^3\right )}{x^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2}{27} \int \left (\frac {37500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) (e x-148)}{\left (-1+3 e^{10/x}\right )^3 x^2}-27 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}+1\right ) (e x-16)+\frac {1800 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (e x^2+10 e x-160\right )}{\left (-1+3 e^{10/x}\right ) x^2}+\frac {375 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (5 e x^2+148 e x-5568\right )}{\left (-1+3 e^{10/x}\right )^2 x^2}-\frac {19062500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right )}{3 \left (-1+3 e^{10/x}\right )^4 x^2}-\frac {7812500 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right )}{3 \left (-1+3 e^{10/x}\right )^5 x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {2}{27} \int \frac {1}{3} \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (-\frac {1500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (-1+3 e^{10/x}\right )^5 x^2}+\frac {9 e \left (108 e^{20/x} x-13 x+3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (-1+3 e^{10/x}\right )^3 x}-81 e^2 x\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{81} \int 3 \exp \left (\frac {\left (54 e^{1+\frac {10}{x}} x-81 e^{1+\frac {20}{x}} x-9 e x+936 e^{10/x}+1296 e^{20/x}+169\right )^2}{81 \left (1-3 e^{10/x}\right )^4}\right ) \left (\frac {500 e^{10/x} \left (13+36 e^{10/x}\right )^3}{\left (1-3 e^{10/x}\right )^5 x^2}+\frac {3 e \left (-108 e^{20/x} x+13 x-3 e^{10/x} (x+500)\right ) \left (13+36 e^{10/x}\right )}{\left (1-3 e^{10/x}\right )^3 x}-27 e^2 x\right )dx\)

Input:

Int[(E^((28561 - 3042*E*x + 81*E^2*x^2 + E^(30/x)*(2426112 - 11664*E*x - 8 
748*E^2*x^2) + E^(10/x)*(316368 + 1404*E*x - 972*E^2*x^2) + E^(20/x)*(1314 
144 + 50382*E*x + 4374*E^2*x^2) + E^(40/x)*(1679616 - 209952*E*x + 6561*E^ 
2*x^2))/(81 - 972*E^(10/x) + 4374*E^(20/x) - 8748*E^(30/x) + 6561*E^(40/x) 
))*(1014*E*x^2 - 54*E^2*x^3 + E^(50/x)*(-209952*E*x^2 + 13122*E^2*x^3) + E 
^(20/x)*(18252000 - 4860*E^2*x^3 + E*(378000*x - 15390*x^2)) + E^(10/x)*(2 
197000 + 810*E^2*x^3 + E*(-117000*x - 3510*x^2)) + E^(30/x)*(50544000 + 14 
580*E^2*x^3 + E*(891000*x + 54270*x^2)) + E^(40/x)*(46656000 - 21870*E^2*x 
^3 + E*(-2916000*x + 58320*x^2))))/(-27*x^2 + 405*E^(10/x)*x^2 - 2430*E^(2 
0/x)*x^2 + 7290*E^(30/x)*x^2 - 10935*E^(40/x)*x^2 + 6561*E^(50/x)*x^2),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(185\) vs. \(2(29)=58\).

Time = 1.53 (sec) , antiderivative size = 186, normalized size of antiderivative = 6.20

\[{\mathrm e}^{\frac {-81 x^{2} {\mathrm e}^{2}-6561 x^{2} {\mathrm e}^{\frac {40+2 x}{x}}+8748 x^{2} {\mathrm e}^{\frac {2 x +30}{x}}-4374 x^{2} {\mathrm e}^{\frac {20+2 x}{x}}+972 x^{2} {\mathrm e}^{\frac {2 x +10}{x}}+3042 x \,{\mathrm e}+209952 x \,{\mathrm e}^{\frac {40+x}{x}}+11664 x \,{\mathrm e}^{\frac {30+x}{x}}-50382 x \,{\mathrm e}^{\frac {20+x}{x}}-1404 x \,{\mathrm e}^{\frac {x +10}{x}}-1679616 \,{\mathrm e}^{\frac {40}{x}}-2426112 \,{\mathrm e}^{\frac {30}{x}}-1314144 \,{\mathrm e}^{\frac {20}{x}}-316368 \,{\mathrm e}^{\frac {10}{x}}-28561}{-6561 \,{\mathrm e}^{\frac {40}{x}}+8748 \,{\mathrm e}^{\frac {30}{x}}-4374 \,{\mathrm e}^{\frac {20}{x}}+972 \,{\mathrm e}^{\frac {10}{x}}-81}}\]

Input:

int(((13122*x^3*exp(1)^2-209952*x^2*exp(1))*exp(5/x)^10+(-21870*x^3*exp(1) 
^2+(58320*x^2-2916000*x)*exp(1)[phone])*exp(5/x)^8+(14580*x^3*exp(1)^2+( 
54270*x^2+891000*x)*exp(1)[phone])*exp(5/x)^6+(-4860*x^3*exp(1)^2+(-1539 
0*x^2+378000*x)*exp(1)[phone])*exp(5/x)^4+(810*x^3*exp(1)^2+(-3510*x^2-1 
17000*x)*exp(1)+2197000)*exp(5/x)^2-54*x^3*exp(1)^2+1014*x^2*exp(1))*exp(( 
(6561*x^2*exp(1)^2-209952*x*exp(1)+1679616)*exp(5/x)^8+(-8748*x^2*exp(1)^2 
-11664*x*exp(1)+2426112)*exp(5/x)^6+(4374*x^2*exp(1)^2+50382*x*exp(1)+1314 
144)*exp(5/x)^4+(-972*x^2*exp(1)^2+1404*x*exp(1)+316368)*exp(5/x)^2+81*x^2 
*exp(1)^2-3042*x*exp(1)+28561)/(6561*exp(5/x)^8-8748*exp(5/x)^6+4374*exp(5 
/x)^4-972*exp(5/x)^2+81))/(6561*x^2*exp(5/x)^10-10935*x^2*exp(5/x)^8+7290* 
x^2*exp(5/x)^6-2430*x^2*exp(5/x)^4+405*x^2*exp(5/x)^2-27*x^2),x)
 

Output:

exp(1/81*(-81*x^2*exp(2)-6561*x^2*exp(2*(20+x)/x)+8748*x^2*exp(2*(x+15)/x) 
-4374*x^2*exp(2*(x+10)/x)+972*x^2*exp(2/x*(5+x))+3042*x*exp(1)+209952*x*ex 
p((40+x)/x)+11664*x*exp((30+x)/x)-50382*x*exp((20+x)/x)-1404*x*exp((x+10)/ 
x)-1679616*exp(40/x)-2426112*exp(30/x)-1314144*exp(20/x)-316368*exp(10/x)- 
28561)/(-81*exp(40/x)+108*exp(30/x)-54*exp(20/x)+12*exp(10/x)-1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (29) = 58\).

Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 4.67 \[ \int \frac {e^{\frac {28561-3042 e x+81 e^2 x^2+e^{30/x} \left (2426112-11664 e x-8748 e^2 x^2\right )+e^{10/x} \left (316368+1404 e x-972 e^2 x^2\right )+e^{20/x} \left (1314144+50382 e x+4374 e^2 x^2\right )+e^{40/x} \left (1679616-209952 e x+6561 e^2 x^2\right )}{81-972 e^{10/x}+4374 e^{20/x}-8748 e^{30/x}+6561 e^{40/x}}} \left (1014 e x^2-54 e^2 x^3+e^{50/x} \left (-209952 e x^2+13122 e^2 x^3\right )+e^{20/x} \left (18252000-4860 e^2 x^3+e \left (378000 x-15390 x^2\right )\right )+e^{10/x} \left (2197000+810 e^2 x^3+e \left (-117000 x-3510 x^2\right )\right )+e^{30/x} \left (50544000+14580 e^2 x^3+e \left (891000 x+54270 x^2\right )\right )+e^{40/x} \left (46656000-21870 e^2 x^3+e \left (-2916000 x+58320 x^2\right )\right )\right )}{-27 x^2+405 e^{10/x} x^2-2430 e^{20/x} x^2+7290 e^{30/x} x^2-10935 e^{40/x} x^2+6561 e^{50/x} x^2} \, dx=e^{\left (\frac {81 \, x^{2} e^{2} - 3042 \, x e + 6561 \, {\left (x^{2} e^{2} - 32 \, x e + 256\right )} e^{\frac {40}{x}} - 2916 \, {\left (3 \, x^{2} e^{2} + 4 \, x e - 832\right )} e^{\frac {30}{x}} + 162 \, {\left (27 \, x^{2} e^{2} + 311 \, x e + 8112\right )} e^{\frac {20}{x}} - 36 \, {\left (27 \, x^{2} e^{2} - 39 \, x e - 8788\right )} e^{\frac {10}{x}} + 28561}{81 \, {\left (81 \, e^{\frac {40}{x}} - 108 \, e^{\frac {30}{x}} + 54 \, e^{\frac {20}{x}} - 12 \, e^{\frac {10}{x}} + 1\right )}}\right )} \] Input:

integrate(((13122*x^3*exp(1)^2-209952*x^2*exp(1))*exp(5/x)^10+(-21870*x^3* 
exp(1)^2+(58320*x^2-2916000*x)*exp(1)[phone])*exp(5/x)^8+(14580*x^3*exp( 
1)^2+(54270*x^2+891000*x)*exp(1)[phone])*exp(5/x)^6+(-4860*x^3*exp(1)^2+ 
(-15390*x^2+378000*x)*exp(1)[phone])*exp(5/x)^4+(810*x^3*exp(1)^2+(-3510 
*x^2-117000*x)*exp(1)+2197000)*exp(5/x)^2-54*x^3*exp(1)^2+1014*x^2*exp(1)) 
*exp(((6561*x^2*exp(1)^2-209952*exp(1)*x+1679616)*exp(5/x)^8+(-8748*x^2*ex 
p(1)^2-11664*exp(1)*x+2426112)*exp(5/x)^6+(4374*x^2*exp(1)^2+50382*exp(1)* 
x+1314144)*exp(5/x)^4+(-972*x^2*exp(1)^2+1404*exp(1)*x+316368)*exp(5/x)^2+ 
81*x^2*exp(1)^2-3042*exp(1)*x+28561)/(6561*exp(5/x)^8-8748*exp(5/x)^6+4374 
*exp(5/x)^4-972*exp(5/x)^2+81))/(6561*x^2*exp(5/x)^10-10935*x^2*exp(5/x)^8 
+7290*x^2*exp(5/x)^6-2430*x^2*exp(5/x)^4+405*x^2*exp(5/x)^2-27*x^2),x, alg 
orithm="fricas")
 

Output:

e^(1/81*(81*x^2*e^2 - 3042*x*e + 6561*(x^2*e^2 - 32*x*e + 256)*e^(40/x) - 
2916*(3*x^2*e^2 + 4*x*e - 832)*e^(30/x) + 162*(27*x^2*e^2 + 311*x*e + 8112 
)*e^(20/x) - 36*(27*x^2*e^2 - 39*x*e - 8788)*e^(10/x) + 28561)/(81*e^(40/x 
) - 108*e^(30/x) + 54*e^(20/x) - 12*e^(10/x) + 1))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (24) = 48\).

Time = 1.70 (sec) , antiderivative size = 134, normalized size of antiderivative = 4.47 \[ \int \frac {e^{\frac {28561-3042 e x+81 e^2 x^2+e^{30/x} \left (2426112-11664 e x-8748 e^2 x^2\right )+e^{10/x} \left (316368+1404 e x-972 e^2 x^2\right )+e^{20/x} \left (1314144+50382 e x+4374 e^2 x^2\right )+e^{40/x} \left (1679616-209952 e x+6561 e^2 x^2\right )}{81-972 e^{10/x}+4374 e^{20/x}-8748 e^{30/x}+6561 e^{40/x}}} \left (1014 e x^2-54 e^2 x^3+e^{50/x} \left (-209952 e x^2+13122 e^2 x^3\right )+e^{20/x} \left (18252000-4860 e^2 x^3+e \left (378000 x-15390 x^2\right )\right )+e^{10/x} \left (2197000+810 e^2 x^3+e \left (-117000 x-3510 x^2\right )\right )+e^{30/x} \left (50544000+14580 e^2 x^3+e \left (891000 x+54270 x^2\right )\right )+e^{40/x} \left (46656000-21870 e^2 x^3+e \left (-2916000 x+58320 x^2\right )\right )\right )}{-27 x^2+405 e^{10/x} x^2-2430 e^{20/x} x^2+7290 e^{30/x} x^2-10935 e^{40/x} x^2+6561 e^{50/x} x^2} \, dx=e^{\frac {81 x^{2} e^{2} - 3042 e x + \left (- 8748 x^{2} e^{2} - 11664 e x + 2426112\right ) e^{\frac {30}{x}} + \left (- 972 x^{2} e^{2} + 1404 e x + 316368\right ) e^{\frac {10}{x}} + \left (4374 x^{2} e^{2} + 50382 e x + 1314144\right ) e^{\frac {20}{x}} + \left (6561 x^{2} e^{2} - 209952 e x + 1679616\right ) e^{\frac {40}{x}} + 28561}{6561 e^{\frac {40}{x}} - 8748 e^{\frac {30}{x}} + 4374 e^{\frac {20}{x}} - 972 e^{\frac {10}{x}} + 81}} \] Input:

integrate(((13122*x**3*exp(1)**2-209952*x**2*exp(1))*exp(5/x)**10+(-21870* 
x**3*exp(1)**2+(58320*x**2-2916000*x)*exp(1)[phone])*exp(5/x)**8+(14580* 
x**3*exp(1)**2+(54270*x**2+891000*x)*exp(1)[phone])*exp(5/x)**6+(-4860*x 
**3*exp(1)**2+(-15390*x**2+378000*x)*exp(1)[phone])*exp(5/x)**4+(810*x** 
3*exp(1)**2+(-3510*x**2-117000*x)*exp(1)+2197000)*exp(5/x)**2-54*x**3*exp( 
1)**2+1014*x**2*exp(1))*exp(((6561*x**2*exp(1)**2-209952*exp(1)*x+1679616) 
*exp(5/x)**8+(-8748*x**2*exp(1)**2-11664*exp(1)*x+2426112)*exp(5/x)**6+(43 
74*x**2*exp(1)**2+50382*exp(1)*x+1314144)*exp(5/x)**4+(-972*x**2*exp(1)**2 
+1404*exp(1)*x+316368)*exp(5/x)**2+81*x**2*exp(1)**2-3042*exp(1)*x+28561)/ 
(6561*exp(5/x)**8-8748*exp(5/x)**6+4374*exp(5/x)**4-972*exp(5/x)**2+81))/( 
6561*x**2*exp(5/x)**10-10935*x**2*exp(5/x)**8+7290*x**2*exp(5/x)**6-2430*x 
**2*exp(5/x)**4+405*x**2*exp(5/x)**2-27*x**2),x)
 

Output:

exp((81*x**2*exp(2) - 3042*E*x + (-8748*x**2*exp(2) - 11664*E*x + 2426112) 
*exp(30/x) + (-972*x**2*exp(2) + 1404*E*x + 316368)*exp(10/x) + (4374*x**2 
*exp(2) + 50382*E*x + 1314144)*exp(20/x) + (6561*x**2*exp(2) - 209952*E*x 
+ 1679616)*exp(40/x) + 28561)/(6561*exp(40/x) - 8748*exp(30/x) + 4374*exp( 
20/x) - 972*exp(10/x) + 81))
 

Maxima [F(-1)]

Timed out. \[ \int \frac {e^{\frac {28561-3042 e x+81 e^2 x^2+e^{30/x} \left (2426112-11664 e x-8748 e^2 x^2\right )+e^{10/x} \left (316368+1404 e x-972 e^2 x^2\right )+e^{20/x} \left (1314144+50382 e x+4374 e^2 x^2\right )+e^{40/x} \left (1679616-209952 e x+6561 e^2 x^2\right )}{81-972 e^{10/x}+4374 e^{20/x}-8748 e^{30/x}+6561 e^{40/x}}} \left (1014 e x^2-54 e^2 x^3+e^{50/x} \left (-209952 e x^2+13122 e^2 x^3\right )+e^{20/x} \left (18252000-4860 e^2 x^3+e \left (378000 x-15390 x^2\right )\right )+e^{10/x} \left (2197000+810 e^2 x^3+e \left (-117000 x-3510 x^2\right )\right )+e^{30/x} \left (50544000+14580 e^2 x^3+e \left (891000 x+54270 x^2\right )\right )+e^{40/x} \left (46656000-21870 e^2 x^3+e \left (-2916000 x+58320 x^2\right )\right )\right )}{-27 x^2+405 e^{10/x} x^2-2430 e^{20/x} x^2+7290 e^{30/x} x^2-10935 e^{40/x} x^2+6561 e^{50/x} x^2} \, dx=\text {Timed out} \] Input:

integrate(((13122*x^3*exp(1)^2-209952*x^2*exp(1))*exp(5/x)^10+(-21870*x^3* 
exp(1)^2+(58320*x^2-2916000*x)*exp(1)[phone])*exp(5/x)^8+(14580*x^3*exp( 
1)^2+(54270*x^2+891000*x)*exp(1)[phone])*exp(5/x)^6+(-4860*x^3*exp(1)^2+ 
(-15390*x^2+378000*x)*exp(1)[phone])*exp(5/x)^4+(810*x^3*exp(1)^2+(-3510 
*x^2-117000*x)*exp(1)+2197000)*exp(5/x)^2-54*x^3*exp(1)^2+1014*x^2*exp(1)) 
*exp(((6561*x^2*exp(1)^2-209952*exp(1)*x+1679616)*exp(5/x)^8+(-8748*x^2*ex 
p(1)^2-11664*exp(1)*x+2426112)*exp(5/x)^6+(4374*x^2*exp(1)^2+50382*exp(1)* 
x+1314144)*exp(5/x)^4+(-972*x^2*exp(1)^2+1404*exp(1)*x+316368)*exp(5/x)^2+ 
81*x^2*exp(1)^2-3042*exp(1)*x+28561)/(6561*exp(5/x)^8-8748*exp(5/x)^6+4374 
*exp(5/x)^4-972*exp(5/x)^2+81))/(6561*x^2*exp(5/x)^10-10935*x^2*exp(5/x)^8 
+7290*x^2*exp(5/x)^6-2430*x^2*exp(5/x)^4+405*x^2*exp(5/x)^2-27*x^2),x, alg 
orithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {e^{\frac {28561-3042 e x+81 e^2 x^2+e^{30/x} \left (2426112-11664 e x-8748 e^2 x^2\right )+e^{10/x} \left (316368+1404 e x-972 e^2 x^2\right )+e^{20/x} \left (1314144+50382 e x+4374 e^2 x^2\right )+e^{40/x} \left (1679616-209952 e x+6561 e^2 x^2\right )}{81-972 e^{10/x}+4374 e^{20/x}-8748 e^{30/x}+6561 e^{40/x}}} \left (1014 e x^2-54 e^2 x^3+e^{50/x} \left (-209952 e x^2+13122 e^2 x^3\right )+e^{20/x} \left (18252000-4860 e^2 x^3+e \left (378000 x-15390 x^2\right )\right )+e^{10/x} \left (2197000+810 e^2 x^3+e \left (-117000 x-3510 x^2\right )\right )+e^{30/x} \left (50544000+14580 e^2 x^3+e \left (891000 x+54270 x^2\right )\right )+e^{40/x} \left (46656000-21870 e^2 x^3+e \left (-2916000 x+58320 x^2\right )\right )\right )}{-27 x^2+405 e^{10/x} x^2-2430 e^{20/x} x^2+7290 e^{30/x} x^2-10935 e^{40/x} x^2+6561 e^{50/x} x^2} \, dx=\int { -\frac {2 \, {\left (27 \, x^{3} e^{2} - 507 \, x^{2} e - 6561 \, {\left (x^{3} e^{2} - 16 \, x^{2} e\right )} e^{\frac {50}{x}} + 3645 \, {\left (3 \, x^{3} e^{2} - 8 \, {\left (x^{2} - 50 \, x\right )} e - 6400\right )} e^{\frac {40}{x}} - 405 \, {\left (18 \, x^{3} e^{2} + {\left (67 \, x^{2} + 1100 \, x\right )} e + 62400\right )} e^{\frac {30}{x}} + 135 \, {\left (18 \, x^{3} e^{2} + {\left (57 \, x^{2} - 1400 \, x\right )} e - 67600\right )} e^{\frac {20}{x}} - 5 \, {\left (81 \, x^{3} e^{2} - 117 \, {\left (3 \, x^{2} + 100 \, x\right )} e + 219700\right )} e^{\frac {10}{x}}\right )} e^{\left (\frac {81 \, x^{2} e^{2} - 3042 \, x e + 6561 \, {\left (x^{2} e^{2} - 32 \, x e + 256\right )} e^{\frac {40}{x}} - 2916 \, {\left (3 \, x^{2} e^{2} + 4 \, x e - 832\right )} e^{\frac {30}{x}} + 162 \, {\left (27 \, x^{2} e^{2} + 311 \, x e + 8112\right )} e^{\frac {20}{x}} - 36 \, {\left (27 \, x^{2} e^{2} - 39 \, x e - 8788\right )} e^{\frac {10}{x}} + 28561}{81 \, {\left (81 \, e^{\frac {40}{x}} - 108 \, e^{\frac {30}{x}} + 54 \, e^{\frac {20}{x}} - 12 \, e^{\frac {10}{x}} + 1\right )}}\right )}}{27 \, {\left (243 \, x^{2} e^{\frac {50}{x}} - 405 \, x^{2} e^{\frac {40}{x}} + 270 \, x^{2} e^{\frac {30}{x}} - 90 \, x^{2} e^{\frac {20}{x}} + 15 \, x^{2} e^{\frac {10}{x}} - x^{2}\right )}} \,d x } \] Input:

integrate(((13122*x^3*exp(1)^2-209952*x^2*exp(1))*exp(5/x)^10+(-21870*x^3* 
exp(1)^2+(58320*x^2-2916000*x)*exp(1)[phone])*exp(5/x)^8+(14580*x^3*exp( 
1)^2+(54270*x^2+891000*x)*exp(1)[phone])*exp(5/x)^6+(-4860*x^3*exp(1)^2+ 
(-15390*x^2+378000*x)*exp(1)[phone])*exp(5/x)^4+(810*x^3*exp(1)^2+(-3510 
*x^2-117000*x)*exp(1)+2197000)*exp(5/x)^2-54*x^3*exp(1)^2+1014*x^2*exp(1)) 
*exp(((6561*x^2*exp(1)^2-209952*exp(1)*x+1679616)*exp(5/x)^8+(-8748*x^2*ex 
p(1)^2-11664*exp(1)*x+2426112)*exp(5/x)^6+(4374*x^2*exp(1)^2+50382*exp(1)* 
x+1314144)*exp(5/x)^4+(-972*x^2*exp(1)^2+1404*exp(1)*x+316368)*exp(5/x)^2+ 
81*x^2*exp(1)^2-3042*exp(1)*x+28561)/(6561*exp(5/x)^8-8748*exp(5/x)^6+4374 
*exp(5/x)^4-972*exp(5/x)^2+81))/(6561*x^2*exp(5/x)^10-10935*x^2*exp(5/x)^8 
+7290*x^2*exp(5/x)^6-2430*x^2*exp(5/x)^4+405*x^2*exp(5/x)^2-27*x^2),x, alg 
orithm="giac")
 

Output:

integrate(-2/27*(27*x^3*e^2 - 507*x^2*e - 6561*(x^3*e^2 - 16*x^2*e)*e^(50/ 
x) + 3645*(3*x^3*e^2 - 8*(x^2 - 50*x)*e - 6400)*e^(40/x) - 405*(18*x^3*e^2 
 + (67*x^2 + 1100*x)*e + 62400)*e^(30/x) + 135*(18*x^3*e^2 + (57*x^2 - 140 
0*x)*e - 67600)*e^(20/x) - 5*(81*x^3*e^2 - 117*(3*x^2 + 100*x)*e + 219700) 
*e^(10/x))*e^(1/81*(81*x^2*e^2 - 3042*x*e + 6561*(x^2*e^2 - 32*x*e + 256)* 
e^(40/x) - 2916*(3*x^2*e^2 + 4*x*e - 832)*e^(30/x) + 162*(27*x^2*e^2 + 311 
*x*e + 8112)*e^(20/x) - 36*(27*x^2*e^2 - 39*x*e - 8788)*e^(10/x) + 28561)/ 
(81*e^(40/x) - 108*e^(30/x) + 54*e^(20/x) - 12*e^(10/x) + 1))/(243*x^2*e^( 
50/x) - 405*x^2*e^(40/x) + 270*x^2*e^(30/x) - 90*x^2*e^(20/x) + 15*x^2*e^( 
10/x) - x^2), x)
 

Mupad [B] (verification not implemented)

Time = 4.55 (sec) , antiderivative size = 697, normalized size of antiderivative = 23.23 \[ \int \frac {e^{\frac {28561-3042 e x+81 e^2 x^2+e^{30/x} \left (2426112-11664 e x-8748 e^2 x^2\right )+e^{10/x} \left (316368+1404 e x-972 e^2 x^2\right )+e^{20/x} \left (1314144+50382 e x+4374 e^2 x^2\right )+e^{40/x} \left (1679616-209952 e x+6561 e^2 x^2\right )}{81-972 e^{10/x}+4374 e^{20/x}-8748 e^{30/x}+6561 e^{40/x}}} \left (1014 e x^2-54 e^2 x^3+e^{50/x} \left (-209952 e x^2+13122 e^2 x^3\right )+e^{20/x} \left (18252000-4860 e^2 x^3+e \left (378000 x-15390 x^2\right )\right )+e^{10/x} \left (2197000+810 e^2 x^3+e \left (-117000 x-3510 x^2\right )\right )+e^{30/x} \left (50544000+14580 e^2 x^3+e \left (891000 x+54270 x^2\right )\right )+e^{40/x} \left (46656000-21870 e^2 x^3+e \left (-2916000 x+58320 x^2\right )\right )\right )}{-27 x^2+405 e^{10/x} x^2-2430 e^{20/x} x^2+7290 e^{30/x} x^2-10935 e^{40/x} x^2+6561 e^{50/x} x^2} \, dx =\text {Too large to display} \] Input:

int((exp((exp(10/x)*(1404*x*exp(1) - 972*x^2*exp(2) + 316368) + exp(20/x)* 
(50382*x*exp(1) + 4374*x^2*exp(2) + 1314144) + exp(40/x)*(6561*x^2*exp(2) 
- 209952*x*exp(1) + 1679616) - exp(30/x)*(11664*x*exp(1) + 8748*x^2*exp(2) 
 - 2426112) - 3042*x*exp(1) + 81*x^2*exp(2) + 28561)/(4374*exp(20/x) - 972 
*exp(10/x) - 8748*exp(30/x) + 6561*exp(40/x) + 81))*(exp(10/x)*(810*x^3*ex 
p(2) - exp(1)*(117000*x + 3510*x^2) + 2197000) + exp(20/x)*(exp(1)*(378000 
*x - 15390*x^2) - 4860*x^3*exp(2) + 18252000) - exp(40/x)*(exp(1)*(2916000 
*x - 58320*x^2) + 21870*x^3*exp(2) - 46656000) + exp(30/x)*(exp(1)*(891000 
*x + 54270*x^2) + 14580*x^3*exp(2) + 50544000) - exp(50/x)*(209952*x^2*exp 
(1) - 13122*x^3*exp(2)) + 1014*x^2*exp(1) - 54*x^3*exp(2)))/(405*x^2*exp(1 
0/x) - 2430*x^2*exp(20/x) + 7290*x^2*exp(30/x) - 10935*x^2*exp(40/x) + 656 
1*x^2*exp(50/x) - 27*x^2),x)
 

Output:

exp((16224*exp(20/x))/(54*exp(20/x) - 12*exp(10/x) - 108*exp(30/x) + 81*ex 
p(40/x) + 1))*exp((20736*exp(40/x))/(54*exp(20/x) - 12*exp(10/x) - 108*exp 
(30/x) + 81*exp(40/x) + 1))*exp((29952*exp(30/x))/(54*exp(20/x) - 12*exp(1 
0/x) - 108*exp(30/x) + 81*exp(40/x) + 1))*exp((35152*exp(10/x))/(486*exp(2 
0/x) - 108*exp(10/x) - 972*exp(30/x) + 729*exp(40/x) + 9))*exp(-(12*x^2*ex 
p(2)*exp(10/x))/(54*exp(20/x) - 12*exp(10/x) - 108*exp(30/x) + 81*exp(40/x 
) + 1))*exp((54*x^2*exp(2)*exp(20/x))/(54*exp(20/x) - 12*exp(10/x) - 108*e 
xp(30/x) + 81*exp(40/x) + 1))*exp((81*x^2*exp(2)*exp(40/x))/(54*exp(20/x) 
- 12*exp(10/x) - 108*exp(30/x) + 81*exp(40/x) + 1))*exp(-(108*x^2*exp(2)*e 
xp(30/x))/(54*exp(20/x) - 12*exp(10/x) - 108*exp(30/x) + 81*exp(40/x) + 1) 
)*exp(-(338*x*exp(1))/(486*exp(20/x) - 108*exp(10/x) - 972*exp(30/x) + 729 
*exp(40/x) + 9))*exp(28561/(4374*exp(20/x) - 972*exp(10/x) - 8748*exp(30/x 
) + 6561*exp(40/x) + 81))*exp((x^2*exp(2))/(54*exp(20/x) - 12*exp(10/x) - 
108*exp(30/x) + 81*exp(40/x) + 1))*exp(-(144*x*exp(1)*exp(30/x))/(54*exp(2 
0/x) - 12*exp(10/x) - 108*exp(30/x) + 81*exp(40/x) + 1))*exp((52*x*exp(1)* 
exp(10/x))/(162*exp(20/x) - 36*exp(10/x) - 324*exp(30/x) + 243*exp(40/x) + 
 3))*exp((622*x*exp(1)*exp(20/x))/(54*exp(20/x) - 12*exp(10/x) - 108*exp(3 
0/x) + 81*exp(40/x) + 1))*exp(-(2592*x*exp(1)*exp(40/x))/(54*exp(20/x) - 1 
2*exp(10/x) - 108*exp(30/x) + 81*exp(40/x) + 1))
 

Reduce [F]

\[ \int \frac {e^{\frac {28561-3042 e x+81 e^2 x^2+e^{30/x} \left (2426112-11664 e x-8748 e^2 x^2\right )+e^{10/x} \left (316368+1404 e x-972 e^2 x^2\right )+e^{20/x} \left (1314144+50382 e x+4374 e^2 x^2\right )+e^{40/x} \left (1679616-209952 e x+6561 e^2 x^2\right )}{81-972 e^{10/x}+4374 e^{20/x}-8748 e^{30/x}+6561 e^{40/x}}} \left (1014 e x^2-54 e^2 x^3+e^{50/x} \left (-209952 e x^2+13122 e^2 x^3\right )+e^{20/x} \left (18252000-4860 e^2 x^3+e \left (378000 x-15390 x^2\right )\right )+e^{10/x} \left (2197000+810 e^2 x^3+e \left (-117000 x-3510 x^2\right )\right )+e^{30/x} \left (50544000+14580 e^2 x^3+e \left (891000 x+54270 x^2\right )\right )+e^{40/x} \left (46656000-21870 e^2 x^3+e \left (-2916000 x+58320 x^2\right )\right )\right )}{-27 x^2+405 e^{10/x} x^2-2430 e^{20/x} x^2+7290 e^{30/x} x^2-10935 e^{40/x} x^2+6561 e^{50/x} x^2} \, dx=\int \frac {\left (\left (13122 x^{3} \left ({\mathrm e}\right )^{2}-209952 x^{2} {\mathrm e}\right ) \left ({\mathrm e}^{\frac {5}{x}}\right )^{10}+\left (-21870 x^{3} \left ({\mathrm e}\right )^{2}+\left (58320 x^{2}-2916000 x \right ) {\mathrm e}[phone]\right ) \left ({\mathrm e}^{\frac {5}{x}}\right )^{8}+\left (14580 x^{3} \left ({\mathrm e}\right )^{2}+\left (54270 x^{2}+891000 x \right ) {\mathrm e}[phone]\right ) \left ({\mathrm e}^{\frac {5}{x}}\right )^{6}+\left (-4860 x^{3} \left ({\mathrm e}\right )^{2}+\left (-15390 x^{2}+378000 x \right ) {\mathrm e}[phone]\right ) \left ({\mathrm e}^{\frac {5}{x}}\right )^{4}+\left (810 x^{3} \left ({\mathrm e}\right )^{2}+\left (-3510 x^{2}-117000 x \right ) {\mathrm e}+2197000\right ) \left ({\mathrm e}^{\frac {5}{x}}\right )^{2}-54 x^{3} \left ({\mathrm e}\right )^{2}+1014 x^{2} {\mathrm e}\right ) {\mathrm e}^{\frac {\left (6561 x^{2} \left ({\mathrm e}\right )^{2}-209952 \,{\mathrm e} x +1679616\right ) \left ({\mathrm e}^{\frac {5}{x}}\right )^{8}+\left (-8748 x^{2} \left ({\mathrm e}\right )^{2}-11664 \,{\mathrm e} x +2426112\right ) \left ({\mathrm e}^{\frac {5}{x}}\right )^{6}+\left (4374 x^{2} \left ({\mathrm e}\right )^{2}+50382 \,{\mathrm e} x +1314144\right ) \left ({\mathrm e}^{\frac {5}{x}}\right )^{4}+\left (-972 x^{2} \left ({\mathrm e}\right )^{2}+1404 \,{\mathrm e} x +316368\right ) \left ({\mathrm e}^{\frac {5}{x}}\right )^{2}+81 x^{2} \left ({\mathrm e}\right )^{2}-3042 \,{\mathrm e} x +28561}{6561 \left ({\mathrm e}^{\frac {5}{x}}\right )^{8}-8748 \left ({\mathrm e}^{\frac {5}{x}}\right )^{6}+4374 \left ({\mathrm e}^{\frac {5}{x}}\right )^{4}-972 \left ({\mathrm e}^{\frac {5}{x}}\right )^{2}+81}}}{6561 x^{2} \left ({\mathrm e}^{\frac {5}{x}}\right )^{10}-10935 x^{2} \left ({\mathrm e}^{\frac {5}{x}}\right )^{8}+7290 x^{2} \left ({\mathrm e}^{\frac {5}{x}}\right )^{6}-2430 x^{2} \left ({\mathrm e}^{\frac {5}{x}}\right )^{4}+405 x^{2} \left ({\mathrm e}^{\frac {5}{x}}\right )^{2}-27 x^{2}}d x \] Input:

int(((13122*x^3*exp(1)^2-209952*x^2*exp(1))*exp(5/x)^10+(-21870*x^3*exp(1) 
^2+(58320*x^2-2916000*x)*exp(1)[phone])*exp(5/x)^8+(14580*x^3*exp(1)^2+( 
54270*x^2+891000*x)*exp(1)[phone])*exp(5/x)^6+(-4860*x^3*exp(1)^2+(-1539 
0*x^2+378000*x)*exp(1)[phone])*exp(5/x)^4+(810*x^3*exp(1)^2+(-3510*x^2-1 
17000*x)*exp(1)+2197000)*exp(5/x)^2-54*x^3*exp(1)^2+1014*x^2*exp(1))*exp(( 
(6561*x^2*exp(1)^2-209952*exp(1)*x+1679616)*exp(5/x)^8+(-8748*x^2*exp(1)^2 
-11664*exp(1)*x+2426112)*exp(5/x)^6+(4374*x^2*exp(1)^2+50382*exp(1)*x+1314 
144)*exp(5/x)^4+(-972*x^2*exp(1)^2+1404*exp(1)*x+316368)*exp(5/x)^2+81*x^2 
*exp(1)^2-3042*exp(1)*x+28561)/(6561*exp(5/x)^8-8748*exp(5/x)^6+4374*exp(5 
/x)^4-972*exp(5/x)^2+81))/(6561*x^2*exp(5/x)^10-10935*x^2*exp(5/x)^8+7290* 
x^2*exp(5/x)^6-2430*x^2*exp(5/x)^4+405*x^2*exp(5/x)^2-27*x^2),x)
 

Output:

int(((13122*x^3*exp(1)^2-209952*x^2*exp(1))*exp(5/x)^10+(-21870*x^3*exp(1) 
^2+(58320*x^2-2916000*x)*exp(1)[phone])*exp(5/x)^8+(14580*x^3*exp(1)^2+( 
54270*x^2+891000*x)*exp(1)[phone])*exp(5/x)^6+(-4860*x^3*exp(1)^2+(-1539 
0*x^2+378000*x)*exp(1)[phone])*exp(5/x)^4+(810*x^3*exp(1)^2+(-3510*x^2-1 
17000*x)*exp(1)+2197000)*exp(5/x)^2-54*x^3*exp(1)^2+1014*x^2*exp(1))*exp(( 
(6561*x^2*exp(1)^2-209952*exp(1)*x+1679616)*exp(5/x)^8+(-8748*x^2*exp(1)^2 
-11664*exp(1)*x+2426112)*exp(5/x)^6+(4374*x^2*exp(1)^2+50382*exp(1)*x+1314 
144)*exp(5/x)^4+(-972*x^2*exp(1)^2+1404*exp(1)*x+316368)*exp(5/x)^2+81*x^2 
*exp(1)^2-3042*exp(1)*x+28561)/(6561*exp(5/x)^8-8748*exp(5/x)^6+4374*exp(5 
/x)^4-972*exp(5/x)^2+81))/(6561*x^2*exp(5/x)^10-10935*x^2*exp(5/x)^8+7290* 
x^2*exp(5/x)^6-2430*x^2*exp(5/x)^4+405*x^2*exp(5/x)^2-27*x^2),x)