Integrand size = 150, antiderivative size = 30 \[ \int e^{-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )} \left (524880 x+574128 x^2+252320 x^3+48160 x^4+3920 x^5+112 x^6+e^{2 x} \left (80 x+48 x^2+32 x^3\right )+e^x \left (-12960 x-10976 x^2-5312 x^3-800 x^4-32 x^5\right )\right ) \, dx=e^{16 e^{-5/x} x^3 \left (-e^x+2 x+(9+x)^2\right )^2} \] Output:
exp(x^3*(8*x-4*exp(x)+4*(x+9)^2)^2/exp(5/x))
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int e^{-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )} \left (524880 x+574128 x^2+252320 x^3+48160 x^4+3920 x^5+112 x^6+e^{2 x} \left (80 x+48 x^2+32 x^3\right )+e^x \left (-12960 x-10976 x^2-5312 x^3-800 x^4-32 x^5\right )\right ) \, dx=e^{16 e^{-5/x} x^3 \left (81-e^x+20 x+x^2\right )^2} \] Input:
Integrate[E^(-5/x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^4 - 32*x^5))/E^(5/x))*(524880*x + 574128*x^2 + 252320*x^3 + 48160*x^4 + 3920*x^5 + 112*x^6 + E^(2*x)*(80*x + 48*x^2 + 32*x^3) + E^x*(-12960*x - 10976*x^2 - 5312*x^3 - 800*x^4 - 32* x^5)),x]
Output:
E^((16*x^3*(81 - E^x + 20*x + x^2)^2)/E^(5/x))
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.
| \(\displaystyle \int \left (112 x^6+3920 x^5+48160 x^4+252320 x^3+574128 x^2+e^{2 x} \left (32 x^3+48 x^2+80 x\right )+e^x \left (-32 x^5-800 x^4-5312 x^3-10976 x^2-12960 x\right )+524880 x\right ) \exp \left (e^{-5/x} \left (16 x^7+640 x^6+8992 x^5+51840 x^4+16 e^{2 x} x^3+104976 x^3+e^x \left (-32 x^5-640 x^4-2592 x^3\right )\right )-\frac {5}{x}\right ) \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (112 x^6 \exp \left (e^{-5/x} \left (16 x^7+640 x^6+8992 x^5+51840 x^4+16 e^{2 x} x^3+104976 x^3+e^x \left (-32 x^5-640 x^4-2592 x^3\right )\right )-\frac {5}{x}\right )+3920 x^5 \exp \left (e^{-5/x} \left (16 x^7+640 x^6+8992 x^5+51840 x^4+16 e^{2 x} x^3+104976 x^3+e^x \left (-32 x^5-640 x^4-2592 x^3\right )\right )-\frac {5}{x}\right )+48160 x^4 \exp \left (e^{-5/x} \left (16 x^7+640 x^6+8992 x^5+51840 x^4+16 e^{2 x} x^3+104976 x^3+e^x \left (-32 x^5-640 x^4-2592 x^3\right )\right )-\frac {5}{x}\right )+252320 x^3 \exp \left (e^{-5/x} \left (16 x^7+640 x^6+8992 x^5+51840 x^4+16 e^{2 x} x^3+104976 x^3+e^x \left (-32 x^5-640 x^4-2592 x^3\right )\right )-\frac {5}{x}\right )+524880 x \exp \left (e^{-5/x} \left (16 x^7+640 x^6+8992 x^5+51840 x^4+16 e^{2 x} x^3+104976 x^3+e^x \left (-32 x^5-640 x^4-2592 x^3\right )\right )-\frac {5}{x}\right )+574128 x^2 \exp \left (e^{-5/x} \left (16 x^7+640 x^6+8992 x^5+51840 x^4+16 e^{2 x} x^3+104976 x^3+e^x \left (-32 x^5-640 x^4-2592 x^3\right )\right )-\frac {5}{x}\right )+16 \left (2 x^2+3 x+5\right ) x \exp \left (e^{-5/x} \left (16 x^7+640 x^6+8992 x^5+51840 x^4+16 e^{2 x} x^3+104976 x^3+e^x \left (-32 x^5-640 x^4-2592 x^3\right )\right )+2 x-\frac {5}{x}\right )-32 \left (x^4+25 x^3+166 x^2+343 x+405\right ) x \exp \left (e^{-5/x} \left (16 x^7+640 x^6+8992 x^5+51840 x^4+16 e^{2 x} x^3+104976 x^3+e^x \left (-32 x^5-640 x^4-2592 x^3\right )\right )+x-\frac {5}{x}\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 16 x \left (7 x^5+245 x^4+3010 x^3+15770 x^2+e^{2 x} \left (2 x^2+3 x+5\right )-2 e^x \left (x^4+25 x^3+166 x^2+343 x+405\right )+35883 x+32805\right ) \exp \left (\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 16 \int \exp \left (-\frac {e^{-5/x} \left (-16 e^{2 x} x^4-16 \left (x^2+20 x+81\right )^2 x^4+32 e^x \left (x^2+20 x+81\right ) x^4+5 e^{5/x}\right )}{x}\right ) x \left (7 x^5+245 x^4+3010 x^3+15770 x^2+35883 x+e^{2 x} \left (2 x^2+3 x+5\right )-2 e^x \left (x^4+25 x^3+166 x^2+343 x+405\right )+32805\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 16 \int \left (\exp \left (2 x-\frac {e^{-5/x} \left (-16 e^{2 x} x^4-16 \left (x^2+20 x+81\right )^2 x^4+32 e^x \left (x^2+20 x+81\right ) x^4+5 e^{5/x}\right )}{x}\right ) x \left (2 x^2+3 x+5\right )-2 \exp \left (x-\frac {e^{-5/x} \left (-16 e^{2 x} x^4-16 \left (x^2+20 x+81\right )^2 x^4+32 e^x \left (x^2+20 x+81\right ) x^4+5 e^{5/x}\right )}{x}\right ) x \left (x^4+25 x^3+166 x^2+343 x+405\right )+\exp \left (-\frac {e^{-5/x} \left (-16 e^{2 x} x^4-16 \left (x^2+20 x+81\right )^2 x^4+32 e^x \left (x^2+20 x+81\right ) x^4+5 e^{5/x}\right )}{x}\right ) x \left (7 x^5+245 x^4+3010 x^3+15770 x^2+35883 x+32805\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 16 \int \exp \left (\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}\right ) x \left (7 x^5+245 x^4+3010 x^3+15770 x^2+35883 x+e^{2 x} \left (2 x^2+3 x+5\right )-2 e^x \left (x^4+25 x^3+166 x^2+343 x+405\right )+32805\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 16 \int \left (\exp \left (2 x+\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}\right ) x \left (2 x^2+3 x+5\right )-2 \exp \left (x+\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}\right ) x \left (x^4+25 x^3+166 x^2+343 x+405\right )+\exp \left (\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}\right ) x \left (7 x^5+245 x^4+3010 x^3+15770 x^2+35883 x+32805\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 16 \int \exp \left (\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}\right ) x \left (7 x^5+245 x^4+3010 x^3+15770 x^2+35883 x+e^{2 x} \left (2 x^2+3 x+5\right )-2 e^x \left (x^4+25 x^3+166 x^2+343 x+405\right )+32805\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 16 \int \left (\exp \left (2 x+\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}\right ) x \left (2 x^2+3 x+5\right )-2 \exp \left (x+\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}\right ) x \left (x^4+25 x^3+166 x^2+343 x+405\right )+\exp \left (\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}\right ) x \left (7 x^5+245 x^4+3010 x^3+15770 x^2+35883 x+32805\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 16 \left (32805 \int e^{\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} xdx-810 \int e^{x+\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} xdx+5 \int e^{2 x+\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} xdx+35883 \int e^{\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} x^2dx-686 \int e^{x+\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} x^2dx+3 \int e^{2 x+\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} x^2dx+15770 \int e^{\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} x^3dx-332 \int e^{x+\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} x^3dx+2 \int e^{2 x+\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} x^3dx+3010 \int e^{\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} x^4dx-50 \int e^{x+\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} x^4dx+245 \int e^{\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} x^5dx-2 \int e^{x+\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} x^5dx+7 \int e^{\frac {e^{-5/x} \left (16 e^{2 x} x^4+16 \left (x^2+20 x+81\right )^2 x^4-32 e^x \left (x^2+20 x+81\right ) x^4-5 e^{5/x}\right )}{x}} x^6dx\right )\) |
Input:
Int[E^(-5/x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^ 6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^4 - 32*x^5))/E^(5/x))*(524880*x + 5741 28*x^2 + 252320*x^3 + 48160*x^4 + 3920*x^5 + 112*x^6 + E^(2*x)*(80*x + 48* x^2 + 32*x^3) + E^x*(-12960*x - 10976*x^2 - 5312*x^3 - 800*x^4 - 32*x^5)), x]
Output:
$Aborted
Time = 124.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77
| method | result | size |
| parallelrisch | \({\mathrm e}^{16 x^{3} \left (x^{4}-2 \,{\mathrm e}^{x} x^{2}+40 x^{3}+{\mathrm e}^{2 x}-40 \,{\mathrm e}^{x} x +562 x^{2}-162 \,{\mathrm e}^{x}+3240 x +6561\right ) {\mathrm e}^{-\frac {5}{x}}}\) | \(53\) |
| risch | \({\mathrm e}^{-16 x^{3} \left (-x^{4}+2 \,{\mathrm e}^{x} x^{2}-40 x^{3}+40 \,{\mathrm e}^{x} x -562 x^{2}+162 \,{\mathrm e}^{x}-{\mathrm e}^{2 x}-3240 x -6561\right ) {\mathrm e}^{-\frac {5}{x}}}\) | \(55\) |
Input:
int(((32*x^3+48*x^2+80*x)*exp(x)^2+(-32*x^5-800*x^4-5312*x^3-10976*x^2-129 60*x)*exp(x)+112*x^6+3920*x^5+48160*x^4+252320*x^3+574128*x^2+524880*x)*ex p((16*exp(x)^2*x^3+(-32*x^5-640*x^4-2592*x^3)*exp(x)+16*x^7+640*x^6+8992*x ^5+51840*x^4+104976*x^3)/exp(5/x))/exp(5/x),x,method=_RETURNVERBOSE)
Output:
exp(16*x^3*(x^4-2*exp(x)*x^2+40*x^3+exp(x)^2-40*exp(x)*x+562*x^2-162*exp(x )+3240*x+6561)/exp(5/x))
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (27) = 54\).
Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.80 \[ \int e^{-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )} \left (524880 x+574128 x^2+252320 x^3+48160 x^4+3920 x^5+112 x^6+e^{2 x} \left (80 x+48 x^2+32 x^3\right )+e^x \left (-12960 x-10976 x^2-5312 x^3-800 x^4-32 x^5\right )\right ) \, dx=e^{\left (\frac {16 \, x^{4} e^{\left (2 \, x - \frac {5}{x}\right )} - 32 \, {\left (x^{6} + 20 \, x^{5} + 81 \, x^{4}\right )} e^{\left (x - \frac {5}{x}\right )} + 16 \, {\left (x^{8} + 40 \, x^{7} + 562 \, x^{6} + 3240 \, x^{5} + 6561 \, x^{4}\right )} e^{\left (-\frac {5}{x}\right )} - 5}{x} + \frac {5}{x}\right )} \] Input:
integrate(((32*x^3+48*x^2+80*x)*exp(x)^2+(-32*x^5-800*x^4-5312*x^3-10976*x ^2-12960*x)*exp(x)+112*x^6+3920*x^5+48160*x^4+252320*x^3+574128*x^2+524880 *x)*exp((16*exp(x)^2*x^3+(-32*x^5-640*x^4-2592*x^3)*exp(x)+16*x^7+640*x^6+ 8992*x^5+51840*x^4+104976*x^3)/exp(5/x))/exp(5/x),x, algorithm="fricas")
Output:
e^((16*x^4*e^(2*x - 5/x) - 32*(x^6 + 20*x^5 + 81*x^4)*e^(x - 5/x) + 16*(x^ 8 + 40*x^7 + 562*x^6 + 3240*x^5 + 6561*x^4)*e^(-5/x) - 5)/x + 5/x)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (26) = 52\).
Time = 15.45 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.03 \[ \int e^{-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )} \left (524880 x+574128 x^2+252320 x^3+48160 x^4+3920 x^5+112 x^6+e^{2 x} \left (80 x+48 x^2+32 x^3\right )+e^x \left (-12960 x-10976 x^2-5312 x^3-800 x^4-32 x^5\right )\right ) \, dx=e^{\left (16 x^{7} + 640 x^{6} + 8992 x^{5} + 51840 x^{4} + 16 x^{3} e^{2 x} + 104976 x^{3} + \left (- 32 x^{5} - 640 x^{4} - 2592 x^{3}\right ) e^{x}\right ) e^{- \frac {5}{x}}} \] Input:
integrate(((32*x**3+48*x**2+80*x)*exp(x)**2+(-32*x**5-800*x**4-5312*x**3-1 0976*x**2-12960*x)*exp(x)+112*x**6+3920*x**5+48160*x**4+252320*x**3+574128 *x**2+524880*x)*exp((16*exp(x)**2*x**3+(-32*x**5-640*x**4-2592*x**3)*exp(x )+16*x**7+640*x**6+8992*x**5+51840*x**4+104976*x**3)/exp(5/x))/exp(5/x),x)
Output:
exp((16*x**7 + 640*x**6 + 8992*x**5 + 51840*x**4 + 16*x**3*exp(2*x) + 1049 76*x**3 + (-32*x**5 - 640*x**4 - 2592*x**3)*exp(x))*exp(-5/x))
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (27) = 54\).
Time = 8.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.70 \[ \int e^{-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )} \left (524880 x+574128 x^2+252320 x^3+48160 x^4+3920 x^5+112 x^6+e^{2 x} \left (80 x+48 x^2+32 x^3\right )+e^x \left (-12960 x-10976 x^2-5312 x^3-800 x^4-32 x^5\right )\right ) \, dx=e^{\left (16 \, x^{7} e^{\left (-\frac {5}{x}\right )} + 640 \, x^{6} e^{\left (-\frac {5}{x}\right )} - 32 \, x^{5} e^{\left (x - \frac {5}{x}\right )} + 8992 \, x^{5} e^{\left (-\frac {5}{x}\right )} - 640 \, x^{4} e^{\left (x - \frac {5}{x}\right )} + 51840 \, x^{4} e^{\left (-\frac {5}{x}\right )} + 16 \, x^{3} e^{\left (2 \, x - \frac {5}{x}\right )} - 2592 \, x^{3} e^{\left (x - \frac {5}{x}\right )} + 104976 \, x^{3} e^{\left (-\frac {5}{x}\right )}\right )} \] Input:
integrate(((32*x^3+48*x^2+80*x)*exp(x)^2+(-32*x^5-800*x^4-5312*x^3-10976*x ^2-12960*x)*exp(x)+112*x^6+3920*x^5+48160*x^4+252320*x^3+574128*x^2+524880 *x)*exp((16*exp(x)^2*x^3+(-32*x^5-640*x^4-2592*x^3)*exp(x)+16*x^7+640*x^6+ 8992*x^5+51840*x^4+104976*x^3)/exp(5/x))/exp(5/x),x, algorithm="maxima")
Output:
e^(16*x^7*e^(-5/x) + 640*x^6*e^(-5/x) - 32*x^5*e^(x - 5/x) + 8992*x^5*e^(- 5/x) - 640*x^4*e^(x - 5/x) + 51840*x^4*e^(-5/x) + 16*x^3*e^(2*x - 5/x) - 2 592*x^3*e^(x - 5/x) + 104976*x^3*e^(-5/x))
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (27) = 54\).
Time = 0.92 (sec) , antiderivative size = 234, normalized size of antiderivative = 7.80 \[ \int e^{-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )} \left (524880 x+574128 x^2+252320 x^3+48160 x^4+3920 x^5+112 x^6+e^{2 x} \left (80 x+48 x^2+32 x^3\right )+e^x \left (-12960 x-10976 x^2-5312 x^3-800 x^4-32 x^5\right )\right ) \, dx=e^{\left (\frac {{\left (16 \, x^{8} e^{\left (\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} + 640 \, x^{7} e^{\left (\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} - 32 \, x^{6} e^{\left (\frac {2 \, x^{2} - 5}{x} + \frac {x^{2} - 5}{x}\right )} + 8992 \, x^{6} e^{\left (\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} - 640 \, x^{5} e^{\left (\frac {2 \, x^{2} - 5}{x} + \frac {x^{2} - 5}{x}\right )} + 51840 \, x^{5} e^{\left (\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} + 16 \, x^{4} e^{\left (\frac {2 \, {\left (2 \, x^{2} - 5\right )}}{x}\right )} - 2592 \, x^{4} e^{\left (\frac {2 \, x^{2} - 5}{x} + \frac {x^{2} - 5}{x}\right )} + 104976 \, x^{4} e^{\left (\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} - 5 \, e^{\left (\frac {2 \, x^{2} - 5}{x}\right )}\right )} e^{\left (-\frac {2 \, x^{2} - 5}{x}\right )}}{x} + \frac {2 \, x^{2} - 5}{x} - \frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} \] Input:
integrate(((32*x^3+48*x^2+80*x)*exp(x)^2+(-32*x^5-800*x^4-5312*x^3-10976*x ^2-12960*x)*exp(x)+112*x^6+3920*x^5+48160*x^4+252320*x^3+574128*x^2+524880 *x)*exp((16*exp(x)^2*x^3+(-32*x^5-640*x^4-2592*x^3)*exp(x)+16*x^7+640*x^6+ 8992*x^5+51840*x^4+104976*x^3)/exp(5/x))/exp(5/x),x, algorithm="giac")
Output:
e^((16*x^8*e^(2*(x^2 - 5)/x) + 640*x^7*e^(2*(x^2 - 5)/x) - 32*x^6*e^((2*x^ 2 - 5)/x + (x^2 - 5)/x) + 8992*x^6*e^(2*(x^2 - 5)/x) - 640*x^5*e^((2*x^2 - 5)/x + (x^2 - 5)/x) + 51840*x^5*e^(2*(x^2 - 5)/x) + 16*x^4*e^(2*(2*x^2 - 5)/x) - 2592*x^4*e^((2*x^2 - 5)/x + (x^2 - 5)/x) + 104976*x^4*e^(2*(x^2 - 5)/x) - 5*e^((2*x^2 - 5)/x))*e^(-(2*x^2 - 5)/x)/x + (2*x^2 - 5)/x - 2*(x^2 - 5)/x)
Time = 2.94 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.97 \[ \int e^{-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )} \left (524880 x+574128 x^2+252320 x^3+48160 x^4+3920 x^5+112 x^6+e^{2 x} \left (80 x+48 x^2+32 x^3\right )+e^x \left (-12960 x-10976 x^2-5312 x^3-800 x^4-32 x^5\right )\right ) \, dx={\mathrm {e}}^{-32\,x^5\,{\mathrm {e}}^{-\frac {5}{x}}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-640\,x^4\,{\mathrm {e}}^{-\frac {5}{x}}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-2592\,x^3\,{\mathrm {e}}^{-\frac {5}{x}}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{16\,x^7\,{\mathrm {e}}^{-\frac {5}{x}}}\,{\mathrm {e}}^{640\,x^6\,{\mathrm {e}}^{-\frac {5}{x}}}\,{\mathrm {e}}^{8992\,x^5\,{\mathrm {e}}^{-\frac {5}{x}}}\,{\mathrm {e}}^{51840\,x^4\,{\mathrm {e}}^{-\frac {5}{x}}}\,{\mathrm {e}}^{104976\,x^3\,{\mathrm {e}}^{-\frac {5}{x}}}\,{\mathrm {e}}^{16\,x^3\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-\frac {5}{x}}} \] Input:
int(exp(exp(-5/x)*(16*x^3*exp(2*x) - exp(x)*(2592*x^3 + 640*x^4 + 32*x^5) + 104976*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7))*exp(-5/x)*(524880 *x + exp(2*x)*(80*x + 48*x^2 + 32*x^3) - exp(x)*(12960*x + 10976*x^2 + 531 2*x^3 + 800*x^4 + 32*x^5) + 574128*x^2 + 252320*x^3 + 48160*x^4 + 3920*x^5 + 112*x^6),x)
Output:
exp(-32*x^5*exp(-5/x)*exp(x))*exp(-640*x^4*exp(-5/x)*exp(x))*exp(-2592*x^3 *exp(-5/x)*exp(x))*exp(16*x^7*exp(-5/x))*exp(640*x^6*exp(-5/x))*exp(8992*x ^5*exp(-5/x))*exp(51840*x^4*exp(-5/x))*exp(104976*x^3*exp(-5/x))*exp(16*x^ 3*exp(2*x)*exp(-5/x))
\[ \int e^{-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )} \left (524880 x+574128 x^2+252320 x^3+48160 x^4+3920 x^5+112 x^6+e^{2 x} \left (80 x+48 x^2+32 x^3\right )+e^x \left (-12960 x-10976 x^2-5312 x^3-800 x^4-32 x^5\right )\right ) \, dx=\int \frac {\left (\left (32 x^{3}+48 x^{2}+80 x \right ) \left ({\mathrm e}^{x}\right )^{2}+\left (-32 x^{5}-800 x^{4}-5312 x^{3}-10976 x^{2}-12960 x \right ) {\mathrm e}^{x}+112 x^{6}+3920 x^{5}+48160 x^{4}+252320 x^{3}+574128 x^{2}+524880 x \right ) {\mathrm e}^{\frac {16 \left ({\mathrm e}^{x}\right )^{2} x^{3}+\left (-32 x^{5}-640 x^{4}-2592 x^{3}\right ) {\mathrm e}^{x}+16 x^{7}+640 x^{6}+8992 x^{5}+51840 x^{4}+104976 x^{3}}{{\mathrm e}^{\frac {5}{x}}}}}{{\mathrm e}^{\frac {5}{x}}}d x \] Input:
int(((32*x^3+48*x^2+80*x)*exp(x)^2+(-32*x^5-800*x^4-5312*x^3-10976*x^2-129 60*x)*exp(x)+112*x^6+3920*x^5+48160*x^4+252320*x^3+574128*x^2+524880*x)*ex p((16*exp(x)^2*x^3+(-32*x^5-640*x^4-2592*x^3)*exp(x)+16*x^7+640*x^6+8992*x ^5+51840*x^4+104976*x^3)/exp(5/x))/exp(5/x),x)
Output:
int(((32*x^3+48*x^2+80*x)*exp(x)^2+(-32*x^5-800*x^4-5312*x^3-10976*x^2-129 60*x)*exp(x)+112*x^6+3920*x^5+48160*x^4+252320*x^3+574128*x^2+524880*x)*ex p((16*exp(x)^2*x^3+(-32*x^5-640*x^4-2592*x^3)*exp(x)+16*x^7+640*x^6+8992*x ^5+51840*x^4+104976*x^3)/exp(5/x))/exp(5/x),x)