\(\int \frac {e^{\frac {216 e^3 x^2+e^2 (-864 x-432 x^2)+e (864+864 x+216 x^2)}{x^2+2 x^3+x^4+e^2 x^4+e (-2 x^3-2 x^4)}} (-432 e^4 x^3+e^2 (-4320 x-5184 x^2-1296 x^3)+e (1728+4320 x+2592 x^2+432 x^3)+e^3 (2592 x^2+1296 x^3))}{-x^3-3 x^4-3 x^5-x^6+e^3 x^6+e^2 (-3 x^5-3 x^6)+e (3 x^4+6 x^5+3 x^6)} \, dx\) [2658]

Optimal result

Integrand size = 194, antiderivative size = 21 \[ \int \frac {e^{\frac {216 e^3 x^2+e^2 \left (-864 x-432 x^2\right )+e \left (864+864 x+216 x^2\right )}{x^2+2 x^3+x^4+e^2 x^4+e \left (-2 x^3-2 x^4\right )}} \left (-432 e^4 x^3+e^2 \left (-4320 x-5184 x^2-1296 x^3\right )+e \left (1728+4320 x+2592 x^2+432 x^3\right )+e^3 \left (2592 x^2+1296 x^3\right )\right )}{-x^3-3 x^4-3 x^5-x^6+e^3 x^6+e^2 \left (-3 x^5-3 x^6\right )+e \left (3 x^4+6 x^5+3 x^6\right )} \, dx=e^{\frac {216 e}{\left (x+\frac {x}{-2-x+e x}\right )^2}} \] Output:

exp(72*exp(1)/(x+x/(x*exp(1)-2-x))/(1/3*x+1/3*x/(x*exp(1)-2-x)))
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {e^{\frac {216 e^3 x^2+e^2 \left (-864 x-432 x^2\right )+e \left (864+864 x+216 x^2\right )}{x^2+2 x^3+x^4+e^2 x^4+e \left (-2 x^3-2 x^4\right )}} \left (-432 e^4 x^3+e^2 \left (-4320 x-5184 x^2-1296 x^3\right )+e \left (1728+4320 x+2592 x^2+432 x^3\right )+e^3 \left (2592 x^2+1296 x^3\right )\right )}{-x^3-3 x^4-3 x^5-x^6+e^3 x^6+e^2 \left (-3 x^5-3 x^6\right )+e \left (3 x^4+6 x^5+3 x^6\right )} \, dx=e^{\frac {216 e (-2+(-1+e) x)^2}{x^2 (-1+(-1+e) x)^2}} \] Input:

Integrate[(E^((216*E^3*x^2 + E^2*(-864*x - 432*x^2) + E*(864 + 864*x + 216 
*x^2))/(x^2 + 2*x^3 + x^4 + E^2*x^4 + E*(-2*x^3 - 2*x^4)))*(-432*E^4*x^3 + 
 E^2*(-4320*x - 5184*x^2 - 1296*x^3) + E*(1728 + 4320*x + 2592*x^2 + 432*x 
^3) + E^3*(2592*x^2 + 1296*x^3)))/(-x^3 - 3*x^4 - 3*x^5 - x^6 + E^3*x^6 + 
E^2*(-3*x^5 - 3*x^6) + E*(3*x^4 + 6*x^5 + 3*x^6)),x]
 

Output:

E^((216*E*(-2 + (-1 + E)*x)^2)/(x^2*(-1 + (-1 + E)*x)^2))
 

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.

\(\displaystyle \int \frac {\left (-432 e^4 x^3+e^2 \left (-1296 x^3-5184 x^2-4320 x\right )+e \left (432 x^3+2592 x^2+4320 x+1728\right )+e^3 \left (1296 x^3+2592 x^2\right )\right ) \exp \left (\frac {216 e^3 x^2+e^2 \left (-432 x^2-864 x\right )+e \left (216 x^2+864 x+864\right )}{e^2 x^4+x^4+2 x^3+x^2+e \left (-2 x^4-2 x^3\right )}\right )}{e^3 x^6-x^6-3 x^5-3 x^4-x^3+e^2 \left (-3 x^6-3 x^5\right )+e \left (3 x^6+6 x^5+3 x^4\right )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (-432 e^4 x^3+e^2 \left (-1296 x^3-5184 x^2-4320 x\right )+e \left (432 x^3+2592 x^2+4320 x+1728\right )+e^3 \left (1296 x^3+2592 x^2\right )\right ) \exp \left (\frac {216 e^3 x^2+e^2 \left (-432 x^2-864 x\right )+e \left (216 x^2+864 x+864\right )}{e^2 x^4+x^4+2 x^3+x^2+e \left (-2 x^4-2 x^3\right )}\right )}{\left (e^3-1\right ) x^6-3 x^5-3 x^4-x^3+e^2 \left (-3 x^6-3 x^5\right )+e \left (3 x^6+6 x^5+3 x^4\right )}dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (-432 e^4 x^3+e^2 \left (-1296 x^3-5184 x^2-4320 x\right )+e \left (432 x^3+2592 x^2+4320 x+1728\right )+e^3 \left (1296 x^3+2592 x^2\right )\right ) \exp \left (\frac {216 e^3 x^2+e^2 \left (-432 x^2-864 x\right )+e \left (216 x^2+864 x+864\right )}{e^2 x^4+x^4+2 x^3+x^2+e \left (-2 x^4-2 x^3\right )}\right )}{x^3 \left (-(1-e)^3 x^3-3 (1-e)^2 x^2-3 (1-e) x-1\right )}dx\)

\(\Big \downarrow \) 2007

\(\displaystyle \int \frac {\left (-432 e^4 x^3+e^2 \left (-1296 x^3-5184 x^2-4320 x\right )+e \left (432 x^3+2592 x^2+4320 x+1728\right )+e^3 \left (1296 x^3+2592 x^2\right )\right ) \exp \left (\frac {216 e^3 x^2+e^2 \left (-432 x^2-864 x\right )+e \left (216 x^2+864 x+864\right )}{e^2 x^4+x^4+2 x^3+x^2+e \left (-2 x^4-2 x^3\right )}\right )}{x^3 ((e-1) x-1)^3}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-432 (1-e)^3 e x^3-2592 (1-e)^2 e x^2-4320 (1-e) e x-1728 e\right ) \exp \left (\frac {216 (1-e)^2 e x^2+864 (1-e) e x+864 e}{x^2 \left ((e-1)^2 x^2-2 (e-1) x+1\right )}\right )}{x^3 (1-(e-1) x)^3}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-432 (1-e)^3 e x^3-2592 (1-e)^2 e x^2-4320 (1-e) e x-1728 e\right ) \exp \left (\frac {216 (1-e)^2 e x^2+864 (1-e) e x+864 e}{x^2 \left ((1-e)^2 x^2+2 (1-e) x+1\right )}\right )}{x^3 (1-(e-1) x)^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {864 (e-1)^3 \exp \left (\frac {216 (1-e)^2 e x^2+864 (1-e) e x+864 e}{x^2 \left ((1-e)^2 x^2+2 (1-e) x+1\right )}+1\right )}{((1-e) x+1)^2}+\frac {432 (e-1)^3 \exp \left (\frac {216 (1-e)^2 e x^2+864 (1-e) e x+864 e}{x^2 \left ((1-e)^2 x^2+2 (1-e) x+1\right )}+1\right )}{((1-e) x+1)^3}-\frac {864 (e-1) \exp \left (\frac {216 (1-e)^2 e x^2+864 (1-e) e x+864 e}{x^2 \left ((1-e)^2 x^2+2 (1-e) x+1\right )}+1\right )}{x^2}-\frac {1728 \exp \left (\frac {216 (1-e)^2 e x^2+864 (1-e) e x+864 e}{x^2 \left ((1-e)^2 x^2+2 (1-e) x+1\right )}+1\right )}{x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 864 (1-e) \int \frac {\exp \left (\frac {216 e ((-1+e) x-2)^2}{x^2 ((-1+e) x-1)^2}+1\right )}{x^2}dx-432 (1-e)^3 \int \frac {\exp \left (\frac {216 e ((-1+e) x-2)^2}{x^2 ((-1+e) x-1)^2}+1\right )}{((1-e) x+1)^3}dx-864 (1-e)^3 \int \frac {\exp \left (\frac {216 e ((-1+e) x-2)^2}{x^2 ((-1+e) x-1)^2}+1\right )}{((1-e) x+1)^2}dx-1728 \int \frac {\exp \left (\frac {216 e ((-1+e) x-2)^2}{x^2 ((-1+e) x-1)^2}+1\right )}{x^3}dx\)

Input:

Int[(E^((216*E^3*x^2 + E^2*(-864*x - 432*x^2) + E*(864 + 864*x + 216*x^2)) 
/(x^2 + 2*x^3 + x^4 + E^2*x^4 + E*(-2*x^3 - 2*x^4)))*(-432*E^4*x^3 + E^2*( 
-4320*x - 5184*x^2 - 1296*x^3) + E*(1728 + 4320*x + 2592*x^2 + 432*x^3) + 
E^3*(2592*x^2 + 1296*x^3)))/(-x^3 - 3*x^4 - 3*x^5 - x^6 + E^3*x^6 + E^2*(- 
3*x^5 - 3*x^6) + E*(3*x^4 + 6*x^5 + 3*x^6)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.19

method	result	size
gosper	\({\mathrm e}^{\frac {216 \,{\mathrm e} \left (x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}-4 x \,{\mathrm e}+x^{2}+4 x +4\right )}{x^{2} \left (-2 x^{2} {\mathrm e}+x^{2} {\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}+2 x +1\right )}}\)	\(67\)
parallelrisch	\({\mathrm e}^{\frac {216 \,{\mathrm e} \left (x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}-4 x \,{\mathrm e}+x^{2}+4 x +4\right )}{x^{2} \left (-2 x^{2} {\mathrm e}+x^{2} {\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}+2 x +1\right )}}\)	\(67\)
risch	\({\mathrm e}^{-\frac {216 \left (x^{2} {\mathrm e}-2 x^{2} {\mathrm e}^{2}+x^{2} {\mathrm e}^{3}+4 x \,{\mathrm e}-4 \,{\mathrm e}^{2} x +4 \,{\mathrm e}\right )}{x^{2} \left (2 x^{2} {\mathrm e}-x^{2} {\mathrm e}^{2}+2 x \,{\mathrm e}-x^{2}-2 x -1\right )}}\)	\(72\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {216 x^{2} {\mathrm e}^{3}+\left (-432 x^{2}-864 x \right ) {\mathrm e}^{2}+\left (216 x^{2}+864 x +864\right ) {\mathrm e}}{x^{4} {\mathrm e}^{2}+\left (-2 x^{4}-2 x^{3}\right ) {\mathrm e}+x^{4}+2 x^{3}+x^{2}}}+\left (-2 \,{\mathrm e}+2\right ) x^{3} {\mathrm e}^{\frac {216 x^{2} {\mathrm e}^{3}+\left (-432 x^{2}-864 x \right ) {\mathrm e}^{2}+\left (216 x^{2}+864 x +864\right ) {\mathrm e}}{x^{4} {\mathrm e}^{2}+\left (-2 x^{4}-2 x^{3}\right ) {\mathrm e}+x^{4}+2 x^{3}+x^{2}}}+\left ({\mathrm e}^{2}-2 \,{\mathrm e}+1\right ) x^{4} {\mathrm e}^{\frac {216 x^{2} {\mathrm e}^{3}+\left (-432 x^{2}-864 x \right ) {\mathrm e}^{2}+\left (216 x^{2}+864 x +864\right ) {\mathrm e}}{x^{4} {\mathrm e}^{2}+\left (-2 x^{4}-2 x^{3}\right ) {\mathrm e}+x^{4}+2 x^{3}+x^{2}}}}{x^{2} \left (x \,{\mathrm e}-x -1\right )^{2}}\)	\(270\)

Input:

int((-432*x^3*exp(1)^4+(1296*x^3+2592*x^2)*exp(1)^3+(-1296*x^3-5184*x^2-43 
20*x)*exp(1)^2+(432*x^3+2592*x^2+4320*x+1728)*exp(1))*exp((216*x^2*exp(1)^ 
3+(-432*x^2-864*x)*exp(1)^2+(216*x^2+864*x+864)*exp(1))/(x^4*exp(1)^2+(-2* 
x^4-2*x^3)*exp(1)+x^4+2*x^3+x^2))/(x^6*exp(1)^3+(-3*x^6-3*x^5)*exp(1)^2+(3 
*x^6+6*x^5+3*x^4)*exp(1)-x^6-3*x^5-3*x^4-x^3),x,method=_RETURNVERBOSE)
 

Output:

exp(216*exp(1)*(x^2*exp(1)^2-2*x^2*exp(1)-4*x*exp(1)+x^2+4*x+4)/x^2/(x^2*e 
xp(1)^2-2*x^2*exp(1)-2*x*exp(1)+x^2+2*x+1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (22) = 44\).

Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.00 \[ \int \frac {e^{\frac {216 e^3 x^2+e^2 \left (-864 x-432 x^2\right )+e \left (864+864 x+216 x^2\right )}{x^2+2 x^3+x^4+e^2 x^4+e \left (-2 x^3-2 x^4\right )}} \left (-432 e^4 x^3+e^2 \left (-4320 x-5184 x^2-1296 x^3\right )+e \left (1728+4320 x+2592 x^2+432 x^3\right )+e^3 \left (2592 x^2+1296 x^3\right )\right )}{-x^3-3 x^4-3 x^5-x^6+e^3 x^6+e^2 \left (-3 x^5-3 x^6\right )+e \left (3 x^4+6 x^5+3 x^6\right )} \, dx=e^{\left (\frac {216 \, {\left (x^{2} e^{3} - 2 \, {\left (x^{2} + 2 \, x\right )} e^{2} + {\left (x^{2} + 4 \, x + 4\right )} e\right )}}{x^{4} e^{2} + x^{4} + 2 \, x^{3} + x^{2} - 2 \, {\left (x^{4} + x^{3}\right )} e}\right )} \] Input:

integrate((-432*x^3*exp(1)^4+(1296*x^3+2592*x^2)*exp(1)^3+(-1296*x^3-5184* 
x^2-4320*x)*exp(1)^2+(432*x^3+2592*x^2+4320*x+1728)*exp(1))*exp((216*x^2*e 
xp(1)^3+(-432*x^2-864*x)*exp(1)^2+(216*x^2+864*x+864)*exp(1))/(x^4*exp(1)^ 
2+(-2*x^4-2*x^3)*exp(1)+x^4+2*x^3+x^2))/(x^6*exp(1)^3+(-3*x^6-3*x^5)*exp(1 
)^2+(3*x^6+6*x^5+3*x^4)*exp(1)-x^6-3*x^5-3*x^4-x^3),x, algorithm="fricas")
 

Output:

e^(216*(x^2*e^3 - 2*(x^2 + 2*x)*e^2 + (x^2 + 4*x + 4)*e)/(x^4*e^2 + x^4 + 
2*x^3 + x^2 - 2*(x^4 + x^3)*e))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (34) = 68\).

Time = 1.64 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.33 \[ \int \frac {e^{\frac {216 e^3 x^2+e^2 \left (-864 x-432 x^2\right )+e \left (864+864 x+216 x^2\right )}{x^2+2 x^3+x^4+e^2 x^4+e \left (-2 x^3-2 x^4\right )}} \left (-432 e^4 x^3+e^2 \left (-4320 x-5184 x^2-1296 x^3\right )+e \left (1728+4320 x+2592 x^2+432 x^3\right )+e^3 \left (2592 x^2+1296 x^3\right )\right )}{-x^3-3 x^4-3 x^5-x^6+e^3 x^6+e^2 \left (-3 x^5-3 x^6\right )+e \left (3 x^4+6 x^5+3 x^6\right )} \, dx=e^{\frac {216 x^{2} e^{3} + \left (- 432 x^{2} - 864 x\right ) e^{2} + e \left (216 x^{2} + 864 x + 864\right )}{x^{4} + x^{4} e^{2} + 2 x^{3} + x^{2} + e \left (- 2 x^{4} - 2 x^{3}\right )}} \] Input:

integrate((-432*x**3*exp(1)**4+(1296*x**3+2592*x**2)*exp(1)**3+(-1296*x**3 
-5184*x**2-4320*x)*exp(1)**2+(432*x**3+2592*x**2+4320*x+1728)*exp(1))*exp( 
(216*x**2*exp(1)**3+(-432*x**2-864*x)*exp(1)**2+(216*x**2+864*x+864)*exp(1 
))/(x**4*exp(1)**2+(-2*x**4-2*x**3)*exp(1)+x**4+2*x**3+x**2))/(x**6*exp(1) 
**3+(-3*x**6-3*x**5)*exp(1)**2+(3*x**6+6*x**5+3*x**4)*exp(1)-x**6-3*x**5-3 
*x**4-x**3),x)
 

Output:

exp((216*x**2*exp(3) + (-432*x**2 - 864*x)*exp(2) + E*(216*x**2 + 864*x + 
864))/(x**4 + x**4*exp(2) + 2*x**3 + x**2 + E*(-2*x**4 - 2*x**3)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (22) = 44\).

Time = 1.95 (sec) , antiderivative size = 146, normalized size of antiderivative = 6.95 \[ \int \frac {e^{\frac {216 e^3 x^2+e^2 \left (-864 x-432 x^2\right )+e \left (864+864 x+216 x^2\right )}{x^2+2 x^3+x^4+e^2 x^4+e \left (-2 x^3-2 x^4\right )}} \left (-432 e^4 x^3+e^2 \left (-4320 x-5184 x^2-1296 x^3\right )+e \left (1728+4320 x+2592 x^2+432 x^3\right )+e^3 \left (2592 x^2+1296 x^3\right )\right )}{-x^3-3 x^4-3 x^5-x^6+e^3 x^6+e^2 \left (-3 x^5-3 x^6\right )+e \left (3 x^4+6 x^5+3 x^6\right )} \, dx=e^{\left (\frac {216 \, e^{3}}{x^{2} {\left (e^{2} - 2 \, e + 1\right )} - 2 \, x {\left (e - 1\right )} + 1} - \frac {864 \, e^{3}}{x {\left (e - 1\right )} - 1} - \frac {432 \, e^{2}}{x^{2} {\left (e^{2} - 2 \, e + 1\right )} - 2 \, x {\left (e - 1\right )} + 1} + \frac {1728 \, e^{2}}{x {\left (e - 1\right )} - 1} + \frac {864 \, e^{2}}{x} + \frac {216 \, e}{x^{2} {\left (e^{2} - 2 \, e + 1\right )} - 2 \, x {\left (e - 1\right )} + 1} - \frac {864 \, e}{x {\left (e - 1\right )} - 1} - \frac {864 \, e}{x} + \frac {864 \, e}{x^{2}}\right )} \] Input:

integrate((-432*x^3*exp(1)^4+(1296*x^3+2592*x^2)*exp(1)^3+(-1296*x^3-5184* 
x^2-4320*x)*exp(1)^2+(432*x^3+2592*x^2+4320*x+1728)*exp(1))*exp((216*x^2*e 
xp(1)^3+(-432*x^2-864*x)*exp(1)^2+(216*x^2+864*x+864)*exp(1))/(x^4*exp(1)^ 
2+(-2*x^4-2*x^3)*exp(1)+x^4+2*x^3+x^2))/(x^6*exp(1)^3+(-3*x^6-3*x^5)*exp(1 
)^2+(3*x^6+6*x^5+3*x^4)*exp(1)-x^6-3*x^5-3*x^4-x^3),x, algorithm="maxima")
 

Output:

e^(216*e^3/(x^2*(e^2 - 2*e + 1) - 2*x*(e - 1) + 1) - 864*e^3/(x*(e - 1) - 
1) - 432*e^2/(x^2*(e^2 - 2*e + 1) - 2*x*(e - 1) + 1) + 1728*e^2/(x*(e - 1) 
 - 1) + 864*e^2/x + 216*e/(x^2*(e^2 - 2*e + 1) - 2*x*(e - 1) + 1) - 864*e/ 
(x*(e - 1) - 1) - 864*e/x + 864*e/x^2)
 

Giac [F]

\[ \int \frac {e^{\frac {216 e^3 x^2+e^2 \left (-864 x-432 x^2\right )+e \left (864+864 x+216 x^2\right )}{x^2+2 x^3+x^4+e^2 x^4+e \left (-2 x^3-2 x^4\right )}} \left (-432 e^4 x^3+e^2 \left (-4320 x-5184 x^2-1296 x^3\right )+e \left (1728+4320 x+2592 x^2+432 x^3\right )+e^3 \left (2592 x^2+1296 x^3\right )\right )}{-x^3-3 x^4-3 x^5-x^6+e^3 x^6+e^2 \left (-3 x^5-3 x^6\right )+e \left (3 x^4+6 x^5+3 x^6\right )} \, dx=\int { -\frac {432 \, {\left (x^{3} e^{4} - 3 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{3} + {\left (3 \, x^{3} + 12 \, x^{2} + 10 \, x\right )} e^{2} - {\left (x^{3} + 6 \, x^{2} + 10 \, x + 4\right )} e\right )} e^{\left (\frac {216 \, {\left (x^{2} e^{3} - 2 \, {\left (x^{2} + 2 \, x\right )} e^{2} + {\left (x^{2} + 4 \, x + 4\right )} e\right )}}{x^{4} e^{2} + x^{4} + 2 \, x^{3} + x^{2} - 2 \, {\left (x^{4} + x^{3}\right )} e}\right )}}{x^{6} e^{3} - x^{6} - 3 \, x^{5} - 3 \, x^{4} - x^{3} - 3 \, {\left (x^{6} + x^{5}\right )} e^{2} + 3 \, {\left (x^{6} + 2 \, x^{5} + x^{4}\right )} e} \,d x } \] Input:

integrate((-432*x^3*exp(1)^4+(1296*x^3+2592*x^2)*exp(1)^3+(-1296*x^3-5184* 
x^2-4320*x)*exp(1)^2+(432*x^3+2592*x^2+4320*x+1728)*exp(1))*exp((216*x^2*e 
xp(1)^3+(-432*x^2-864*x)*exp(1)^2+(216*x^2+864*x+864)*exp(1))/(x^4*exp(1)^ 
2+(-2*x^4-2*x^3)*exp(1)+x^4+2*x^3+x^2))/(x^6*exp(1)^3+(-3*x^6-3*x^5)*exp(1 
)^2+(3*x^6+6*x^5+3*x^4)*exp(1)-x^6-3*x^5-3*x^4-x^3),x, algorithm="giac")
 

Output:

integrate(-432*(x^3*e^4 - 3*(x^3 + 2*x^2)*e^3 + (3*x^3 + 12*x^2 + 10*x)*e^ 
2 - (x^3 + 6*x^2 + 10*x + 4)*e)*e^(216*(x^2*e^3 - 2*(x^2 + 2*x)*e^2 + (x^2 
 + 4*x + 4)*e)/(x^4*e^2 + x^4 + 2*x^3 + x^2 - 2*(x^4 + x^3)*e))/(x^6*e^3 - 
 x^6 - 3*x^5 - 3*x^4 - x^3 - 3*(x^6 + x^5)*e^2 + 3*(x^6 + 2*x^5 + x^4)*e), 
 x)
 

Mupad [B] (verification not implemented)

Time = 3.07 (sec) , antiderivative size = 213, normalized size of antiderivative = 10.14 \[ \int \frac {e^{\frac {216 e^3 x^2+e^2 \left (-864 x-432 x^2\right )+e \left (864+864 x+216 x^2\right )}{x^2+2 x^3+x^4+e^2 x^4+e \left (-2 x^3-2 x^4\right )}} \left (-432 e^4 x^3+e^2 \left (-4320 x-5184 x^2-1296 x^3\right )+e \left (1728+4320 x+2592 x^2+432 x^3\right )+e^3 \left (2592 x^2+1296 x^3\right )\right )}{-x^3-3 x^4-3 x^5-x^6+e^3 x^6+e^2 \left (-3 x^5-3 x^6\right )+e \left (3 x^4+6 x^5+3 x^6\right )} \, dx={\mathrm {e}}^{\frac {216\,\mathrm {e}}{2\,x-2\,x\,\mathrm {e}-2\,x^2\,\mathrm {e}+x^2\,{\mathrm {e}}^2+x^2+1}}\,{\mathrm {e}}^{\frac {216\,{\mathrm {e}}^3}{2\,x-2\,x\,\mathrm {e}-2\,x^2\,\mathrm {e}+x^2\,{\mathrm {e}}^2+x^2+1}}\,{\mathrm {e}}^{-\frac {432\,{\mathrm {e}}^2}{2\,x-2\,x\,\mathrm {e}-2\,x^2\,\mathrm {e}+x^2\,{\mathrm {e}}^2+x^2+1}}\,{\mathrm {e}}^{\frac {864\,\mathrm {e}}{x-2\,x^2\,\mathrm {e}-2\,x^3\,\mathrm {e}+x^3\,{\mathrm {e}}^2+2\,x^2+x^3}}\,{\mathrm {e}}^{-\frac {864\,{\mathrm {e}}^2}{x-2\,x^2\,\mathrm {e}-2\,x^3\,\mathrm {e}+x^3\,{\mathrm {e}}^2+2\,x^2+x^3}}\,{\mathrm {e}}^{\frac {864\,\mathrm {e}}{x^4\,{\mathrm {e}}^2-2\,x^4\,\mathrm {e}-2\,x^3\,\mathrm {e}+x^2+2\,x^3+x^4}} \] Input:

int(-(exp((exp(1)*(864*x + 216*x^2 + 864) - exp(2)*(864*x + 432*x^2) + 216 
*x^2*exp(3))/(x^4*exp(2) - exp(1)*(2*x^3 + 2*x^4) + x^2 + 2*x^3 + x^4))*(e 
xp(1)*(4320*x + 2592*x^2 + 432*x^3 + 1728) - exp(2)*(4320*x + 5184*x^2 + 1 
296*x^3) + exp(3)*(2592*x^2 + 1296*x^3) - 432*x^3*exp(4)))/(exp(2)*(3*x^5 
+ 3*x^6) - x^6*exp(3) - exp(1)*(3*x^4 + 6*x^5 + 3*x^6) + x^3 + 3*x^4 + 3*x 
^5 + x^6),x)
 

Output:

exp((216*exp(1))/(2*x - 2*x*exp(1) - 2*x^2*exp(1) + x^2*exp(2) + x^2 + 1)) 
*exp((216*exp(3))/(2*x - 2*x*exp(1) - 2*x^2*exp(1) + x^2*exp(2) + x^2 + 1) 
)*exp(-(432*exp(2))/(2*x - 2*x*exp(1) - 2*x^2*exp(1) + x^2*exp(2) + x^2 + 
1))*exp((864*exp(1))/(x - 2*x^2*exp(1) - 2*x^3*exp(1) + x^3*exp(2) + 2*x^2 
 + x^3))*exp(-(864*exp(2))/(x - 2*x^2*exp(1) - 2*x^3*exp(1) + x^3*exp(2) + 
 2*x^2 + x^3))*exp((864*exp(1))/(x^4*exp(2) - 2*x^4*exp(1) - 2*x^3*exp(1) 
+ x^2 + 2*x^3 + x^4))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 5.10 \[ \int \frac {e^{\frac {216 e^3 x^2+e^2 \left (-864 x-432 x^2\right )+e \left (864+864 x+216 x^2\right )}{x^2+2 x^3+x^4+e^2 x^4+e \left (-2 x^3-2 x^4\right )}} \left (-432 e^4 x^3+e^2 \left (-4320 x-5184 x^2-1296 x^3\right )+e \left (1728+4320 x+2592 x^2+432 x^3\right )+e^3 \left (2592 x^2+1296 x^3\right )\right )}{-x^3-3 x^4-3 x^5-x^6+e^3 x^6+e^2 \left (-3 x^5-3 x^6\right )+e \left (3 x^4+6 x^5+3 x^6\right )} \, dx=\frac {e^{\frac {216 e^{3} x^{2}+216 e \,x^{2}+864 e x +864 e}{e^{2} x^{4}-2 e \,x^{4}-2 e \,x^{3}+x^{4}+2 x^{3}+x^{2}}}}{e^{\frac {432 e^{2} x +864 e^{2}}{e^{2} x^{3}-2 e \,x^{3}-2 e \,x^{2}+x^{3}+2 x^{2}+x}}} \] Input:

int((-432*x^3*exp(1)^4+(1296*x^3+2592*x^2)*exp(1)^3+(-1296*x^3-5184*x^2-43 
20*x)*exp(1)^2+(432*x^3+2592*x^2+4320*x+1728)*exp(1))*exp((216*x^2*exp(1)^ 
3+(-432*x^2-864*x)*exp(1)^2+(216*x^2+864*x+864)*exp(1))/(x^4*exp(1)^2+(-2* 
x^4-2*x^3)*exp(1)+x^4+2*x^3+x^2))/(x^6*exp(1)^3+(-3*x^6-3*x^5)*exp(1)^2+(3 
*x^6+6*x^5+3*x^4)*exp(1)-x^6-3*x^5-3*x^4-x^3),x)
 

Output:

e**((216*e**3*x**2 + 216*e*x**2 + 864*e*x + 864*e)/(e**2*x**4 - 2*e*x**4 - 
 2*e*x**3 + x**4 + 2*x**3 + x**2))/e**((432*e**2*x + 864*e**2)/(e**2*x**3 
- 2*e*x**3 - 2*e*x**2 + x**3 + 2*x**2 + x))