\(\int \frac {e^{-\frac {e^3 (-3 x+x^3)}{1+e^3 (36 x-24 x^2+4 x^3)}} (e^3 (3-3 x^2)+e^6 (72 x^2-96 x^3+24 x^4))}{1+e^3 (72 x-48 x^2+8 x^3)+e^6 (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6)} \, dx\) [599]

Optimal result

Integrand size = 121, antiderivative size = 28 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=4+e^{-\frac {-3+x^2}{(6-2 x)^2+\frac {1}{e^3 x}}} \] Output:

1/exp((x^2-3)/(1/x/exp(3)+(6-2*x)^2))+4
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=e^{-\frac {e^3 x \left (-3+x^2\right )}{1+4 e^3 (-3+x)^2 x}} \] Input:

Integrate[(E^3*(3 - 3*x^2) + E^6*(72*x^2 - 96*x^3 + 24*x^4))/(E^((E^3*(-3* 
x + x^3))/(1 + E^3*(36*x - 24*x^2 + 4*x^3)))*(1 + E^3*(72*x - 48*x^2 + 8*x 
^3) + E^6*(1296*x^2 - 1728*x^3 + 864*x^4 - 192*x^5 + 16*x^6))),x]
 

Output:

E^(-((E^3*x*(-3 + x^2))/(1 + 4*E^3*(-3 + x)^2*x)))
 

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.

\(\displaystyle \int \frac {\left (e^3 \left (3-3 x^2\right )+e^6 \left (24 x^4-96 x^3+72 x^2\right )\right ) \exp \left (-\frac {e^3 \left (x^3-3 x\right )}{e^3 \left (4 x^3-24 x^2+36 x\right )+1}\right )}{e^3 \left (8 x^3-48 x^2+72 x\right )+e^6 \left (16 x^6-192 x^5+864 x^4-1728 x^3+1296 x^2\right )+1} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \frac {\left (e^3 \left (3-3 x^2\right )+e^6 \left (24 x^4-96 x^3+72 x^2\right )\right ) \exp \left (-\frac {e^3 \left (x^3-3 x\right )}{e^3 \left (4 x^3-24 x^2+36 x\right )+1}\right )}{\left (4 e^3 x^3-24 e^3 x^2+36 e^3 x+1\right )^2}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (24 e^6 x^4-96 e^6 x^3-3 e^3 \left (1-24 e^3\right ) x^2+3 e^3\right ) \exp \left (-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right )}{\left (4 e^3 x^3-24 e^3 x^2+36 e^3 x+1\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {6 (x+2) \exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}+\frac {3 \left (-\left (\left (1-48 e^3\right ) x^2\right )-2 \left (1+72 e^3\right ) x-3\right ) \exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right )}{\left (4 e^3 x^3-24 e^3 x^2+36 e^3 x+1\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -9 \int \frac {\exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right )}{\left (4 e^3 x^3-24 e^3 x^2+36 e^3 x+1\right )^2}dx-6 \left (1+72 e^3\right ) \int \frac {\exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right ) x}{\left (4 e^3 x^3-24 e^3 x^2+36 e^3 x+1\right )^2}dx-3 \left (1-48 e^3\right ) \int \frac {\exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right ) x^2}{\left (4 e^3 x^3-24 e^3 x^2+36 e^3 x+1\right )^2}dx+12 \int \frac {\exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}dx+6 \int \frac {\exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right ) x}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}dx\)

Input:

Int[(E^3*(3 - 3*x^2) + E^6*(72*x^2 - 96*x^3 + 24*x^4))/(E^((E^3*(-3*x + x^ 
3))/(1 + E^3*(36*x - 24*x^2 + 4*x^3)))*(1 + E^3*(72*x - 48*x^2 + 8*x^3) + 
E^6*(1296*x^2 - 1728*x^3 + 864*x^4 - 192*x^5 + 16*x^6))),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25

method	result	size
norman	\({\mathrm e}^{-\frac {\left (x^{3}-3 x \right ) {\mathrm e}^{3}}{\left (4 x^{3}-24 x^{2}+36 x \right ) {\mathrm e}^{3}+1}}\)	\(35\)
risch	\({\mathrm e}^{-\frac {x \left (x^{2}-3\right ) {\mathrm e}^{3}}{4 x^{3} {\mathrm e}^{3}-24 x^{2} {\mathrm e}^{3}+36 x \,{\mathrm e}^{3}+1}}\)	\(35\)
gosper	\({\mathrm e}^{-\frac {x \left (x^{2}-3\right ) {\mathrm e}^{3}}{4 x^{3} {\mathrm e}^{3}-24 x^{2} {\mathrm e}^{3}+36 x \,{\mathrm e}^{3}+1}}\)	\(36\)
parallelrisch	\({\mathrm e}^{-\frac {\left (x^{3}-3 x \right ) {\mathrm e}^{3}}{4 x^{3} {\mathrm e}^{3}-24 x^{2} {\mathrm e}^{3}+36 x \,{\mathrm e}^{3}+1}}\)	\(37\)

Input:

int(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192*x^5+8 
64*x^4-1728*x^3+1296*x^2)*exp(3)^2+(8*x^3-48*x^2+72*x)*exp(3)+1)/exp((x^3- 
3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x,method=_RETURNVERBOSE)
 

Output:

1/exp((x^3-3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=e^{\left (-\frac {{\left (x^{3} - 3 \, x\right )} e^{3}}{4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{3} + 1}\right )} \] Input:

integrate(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192 
*x^5+864*x^4-1728*x^3+1296*x^2)*exp(3)^2+(8*x^3-48*x^2+72*x)*exp(3)+1)/exp 
((x^3-3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x, algorithm="fricas")
 

Output:

e^(-(x^3 - 3*x)*e^3/(4*(x^3 - 6*x^2 + 9*x)*e^3 + 1))
 

Sympy [A] (verification not implemented)

Time = 0.96 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=e^{- \frac {\left (x^{3} - 3 x\right ) e^{3}}{\left (4 x^{3} - 24 x^{2} + 36 x\right ) e^{3} + 1}} \] Input:

integrate(((24*x**4-96*x**3+72*x**2)*exp(3)**2+(-3*x**2+3)*exp(3))/((16*x* 
*6-192*x**5+864*x**4-1728*x**3+1296*x**2)*exp(3)**2+(8*x**3-48*x**2+72*x)* 
exp(3)+1)/exp((x**3-3*x)*exp(3)/((4*x**3-24*x**2+36*x)*exp(3)+1)),x)
 

Output:

exp(-(x**3 - 3*x)*exp(3)/((4*x**3 - 24*x**2 + 36*x)*exp(3) + 1))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.07 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=e^{\left (-\frac {6 \, x^{2} e^{3}}{4 \, x^{3} e^{3} - 24 \, x^{2} e^{3} + 36 \, x e^{3} + 1} + \frac {12 \, x e^{3}}{4 \, x^{3} e^{3} - 24 \, x^{2} e^{3} + 36 \, x e^{3} + 1} + \frac {1}{4 \, {\left (4 \, x^{3} e^{3} - 24 \, x^{2} e^{3} + 36 \, x e^{3} + 1\right )}} - \frac {1}{4}\right )} \] Input:

integrate(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192 
*x^5+864*x^4-1728*x^3+1296*x^2)*exp(3)^2+(8*x^3-48*x^2+72*x)*exp(3)+1)/exp 
((x^3-3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x, algorithm="maxima")
 

Output:

e^(-6*x^2*e^3/(4*x^3*e^3 - 24*x^2*e^3 + 36*x*e^3 + 1) + 12*x*e^3/(4*x^3*e^ 
3 - 24*x^2*e^3 + 36*x*e^3 + 1) + 1/4/(4*x^3*e^3 - 24*x^2*e^3 + 36*x*e^3 + 
1) - 1/4)
 

Giac [F]

\[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=\int { \frac {3 \, {\left (8 \, {\left (x^{4} - 4 \, x^{3} + 3 \, x^{2}\right )} e^{6} - {\left (x^{2} - 1\right )} e^{3}\right )} e^{\left (-\frac {{\left (x^{3} - 3 \, x\right )} e^{3}}{4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{3} + 1}\right )}}{16 \, {\left (x^{6} - 12 \, x^{5} + 54 \, x^{4} - 108 \, x^{3} + 81 \, x^{2}\right )} e^{6} + 8 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{3} + 1} \,d x } \] Input:

integrate(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192 
*x^5+864*x^4-1728*x^3+1296*x^2)*exp(3)^2+(8*x^3-48*x^2+72*x)*exp(3)+1)/exp 
((x^3-3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x, algorithm="giac")
 

Output:

integrate(3*(8*(x^4 - 4*x^3 + 3*x^2)*e^6 - (x^2 - 1)*e^3)*e^(-(x^3 - 3*x)* 
e^3/(4*(x^3 - 6*x^2 + 9*x)*e^3 + 1))/(16*(x^6 - 12*x^5 + 54*x^4 - 108*x^3 
+ 81*x^2)*e^6 + 8*(x^3 - 6*x^2 + 9*x)*e^3 + 1), x)
 

Mupad [B] (verification not implemented)

Time = 3.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx={\mathrm {e}}^{\frac {3\,x\,{\mathrm {e}}^3-x^3\,{\mathrm {e}}^3}{4\,{\mathrm {e}}^3\,x^3-24\,{\mathrm {e}}^3\,x^2+36\,{\mathrm {e}}^3\,x+1}} \] Input:

int(-(exp((exp(3)*(3*x - x^3))/(exp(3)*(36*x - 24*x^2 + 4*x^3) + 1))*(exp( 
3)*(3*x^2 - 3) - exp(6)*(72*x^2 - 96*x^3 + 24*x^4)))/(exp(6)*(1296*x^2 - 1 
728*x^3 + 864*x^4 - 192*x^5 + 16*x^6) + exp(3)*(72*x - 48*x^2 + 8*x^3) + 1 
),x)
 

Output:

exp((3*x*exp(3) - x^3*exp(3))/(36*x*exp(3) - 24*x^2*exp(3) + 4*x^3*exp(3) 
+ 1))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.57 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=\frac {e^{\frac {3 e^{3} x}{4 e^{3} x^{3}-24 e^{3} x^{2}+36 e^{3} x +1}}}{e^{\frac {e^{3} x^{3}}{4 e^{3} x^{3}-24 e^{3} x^{2}+36 e^{3} x +1}}} \] Input:

int(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192*x^5+8 
64*x^4-1728*x^3+1296*x^2)*exp(3)^2+(8*x^3-48*x^2+72*x)*exp(3)+1)/exp((x^3- 
3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x)
 

Output:

e**((3*e**3*x)/(4*e**3*x**3 - 24*e**3*x**2 + 36*e**3*x + 1))/e**((e**3*x** 
3)/(4*e**3*x**3 - 24*e**3*x**2 + 36*e**3*x + 1))