\(\int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} ((32-8 x-12 e^{4+x} x+16 x^2) \log (x)+(-16+8 x-24 x^2+e^{4+x} (12 x+6 x^2)) \log ^2(x))}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} (-48 x^3+12 x^4-24 x^5)} \, dx\) [28]

Optimal result

Integrand size = 149, antiderivative size = 31 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=e^{\frac {\log ^2(x)}{x \left (4-x+2 x \left (-\frac {3 e^{4+x}}{4}+x\right )\right )}} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=e^{\frac {2 \log ^2(x)}{x \left (8-2 x-3 e^{4+x} x+4 x^2\right )}} \] Input:

Integrate[((32 - 8*x - 12*E^(4 + x)*x + 16*x^2)*Log[x] + (-16 + 8*x - 24*x 
^2 + E^(4 + x)*(12*x + 6*x^2))*Log[x]^2)/(E^((2*Log[x]^2)/(-8*x + 2*x^2 + 
3*E^(4 + x)*x^2 - 4*x^3))*(64*x^2 - 32*x^3 + 68*x^4 + 9*E^(8 + 2*x)*x^4 - 
16*x^5 + 16*x^6 + E^(4 + x)*(-48*x^3 + 12*x^4 - 24*x^5))),x]
 

Output:

E^((2*Log[x]^2)/(x*(8 - 2*x - 3*E^(4 + x)*x + 4*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.

\(\displaystyle \int \frac {\left (\left (-24 x^2+e^{x+4} \left (6 x^2+12 x\right )+8 x-16\right ) \log ^2(x)+\left (16 x^2-12 e^{x+4} x-8 x+32\right ) \log (x)\right ) \exp \left (-\frac {2 \log ^2(x)}{-4 x^3+3 e^{x+4} x^2+2 x^2-8 x}\right )}{16 x^6-16 x^5+9 e^{2 x+8} x^4+68 x^4-32 x^3+64 x^2+e^{x+4} \left (-24 x^5+12 x^4-48 x^3\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (\left (-24 x^2+e^{x+4} \left (6 x^2+12 x\right )+8 x-16\right ) \log ^2(x)+\left (16 x^2-12 e^{x+4} x-8 x+32\right ) \log (x)\right ) \exp \left (-\frac {2 \log ^2(x)}{-4 x^3+3 e^{x+4} x^2+2 x^2-8 x}\right )}{x^2 \left (4 x^2-3 e^{x+4} x-2 x+8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (2 x^3-3 x^2+4 x+4\right ) \log ^2(x) \exp \left (-\frac {2 \log ^2(x)}{-4 x^3+3 e^{x+4} x^2+2 x^2-8 x}\right )}{x^2 \left (4 x^2-3 e^{x+4} x-2 x+8\right )^2}-\frac {2 \log (x) (x \log (x)+2 \log (x)-2) \exp \left (-\frac {2 \log ^2(x)}{-4 x^3+3 e^{x+4} x^2+2 x^2-8 x}\right )}{x^2 \left (4 x^2-3 e^{x+4} x-2 x+8\right )}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (2 x^3-3 x^2+4 x+4\right ) \log ^2(x) \exp \left (\frac {2 \log ^2(x)}{x \left (4 x^2-\left (3 e^{x+4}+2\right ) x+8\right )}\right )}{x^2 \left (4 x^2-3 e^{x+4} x-2 x+8\right )^2}-\frac {2 \log (x) (x \log (x)+2 \log (x)-2) \exp \left (\frac {2 \log ^2(x)}{x \left (4 x^2-\left (3 e^{x+4}+2\right ) x+8\right )}\right )}{x^2 \left (4 x^2-3 e^{x+4} x-2 x+8\right )}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {4 \left (2 x^3-3 x^2+4 x+4\right ) \log ^2(x) \exp \left (\frac {2 \log ^2(x)}{x \left (4 x^2-\left (3 e^{x+4}+2\right ) x+8\right )}\right )}{x^2 \left (4 x^2-3 e^{x+4} x-2 x+8\right )^2}-\frac {2 \log (x) (x \log (x)+2 \log (x)-2) \exp \left (\frac {2 \log ^2(x)}{x \left (4 x^2-\left (3 e^{x+4}+2\right ) x+8\right )}\right )}{x^2 \left (4 x^2-3 e^{x+4} x-2 x+8\right )}\right )dx\)

Input:

Int[((32 - 8*x - 12*E^(4 + x)*x + 16*x^2)*Log[x] + (-16 + 8*x - 24*x^2 + E 
^(4 + x)*(12*x + 6*x^2))*Log[x]^2)/(E^((2*Log[x]^2)/(-8*x + 2*x^2 + 3*E^(4 
 + x)*x^2 - 4*x^3))*(64*x^2 - 32*x^3 + 68*x^4 + 9*E^(8 + 2*x)*x^4 - 16*x^5 
 + 16*x^6 + E^(4 + x)*(-48*x^3 + 12*x^4 - 24*x^5))),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97

\[{\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{x \left (-3 x \,{\mathrm e}^{4+x}+4 x^{2}-2 x +8\right )}}\]

Input:

int((((6*x^2+12*x)*exp(4+x)-24*x^2+8*x-16)*ln(x)^2+(-12*x*exp(4+x)+16*x^2- 
8*x+32)*ln(x))*exp(-2*ln(x)^2/(3*x^2*exp(4+x)-4*x^3+2*x^2-8*x))/(9*x^4*exp 
(4+x)^2+(-24*x^5+12*x^4-48*x^3)*exp(4+x)+16*x^6-16*x^5+68*x^4-32*x^3+64*x^ 
2),x)
 

Output:

exp(2*ln(x)^2/x/(-3*x*exp(4+x)+4*x^2-2*x+8))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=e^{\left (\frac {2 \, \log \left (x\right )^{2}}{4 \, x^{3} - 3 \, x^{2} e^{\left (x + 4\right )} - 2 \, x^{2} + 8 \, x}\right )} \] Input:

integrate((((6*x^2+12*x)*exp(4+x)-24*x^2+8*x-16)*log(x)^2+(-12*x*exp(4+x)+ 
16*x^2-8*x+32)*log(x))*exp(-2*log(x)^2/(3*x^2*exp(4+x)-4*x^3+2*x^2-8*x))/( 
9*x^4*exp(4+x)^2+(-24*x^5+12*x^4-48*x^3)*exp(4+x)+16*x^6-16*x^5+68*x^4-32* 
x^3+64*x^2),x, algorithm="fricas")
 

Output:

e^(2*log(x)^2/(4*x^3 - 3*x^2*e^(x + 4) - 2*x^2 + 8*x))
 

Sympy [A] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=e^{- \frac {2 \log {\left (x \right )}^{2}}{- 4 x^{3} + 3 x^{2} e^{x + 4} + 2 x^{2} - 8 x}} \] Input:

integrate((((6*x**2+12*x)*exp(4+x)-24*x**2+8*x-16)*ln(x)**2+(-12*x*exp(4+x 
)+16*x**2-8*x+32)*ln(x))*exp(-2*ln(x)**2/(3*x**2*exp(4+x)-4*x**3+2*x**2-8* 
x))/(9*x**4*exp(4+x)**2+(-24*x**5+12*x**4-48*x**3)*exp(4+x)+16*x**6-16*x** 
5+68*x**4-32*x**3+64*x**2),x)
 

Output:

exp(-2*log(x)**2/(-4*x**3 + 3*x**2*exp(x + 4) + 2*x**2 - 8*x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (29) = 58\).

Time = 0.35 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.94 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=e^{\left (-\frac {x \log \left (x\right )^{2}}{4 \, x^{2} - 3 \, x e^{\left (x + 4\right )} - 2 \, x + 8} + \frac {3 \, e^{\left (x + 4\right )} \log \left (x\right )^{2}}{4 \, {\left (4 \, x^{2} - 3 \, x e^{\left (x + 4\right )} - 2 \, x + 8\right )}} + \frac {\log \left (x\right )^{2}}{2 \, {\left (4 \, x^{2} - 3 \, x e^{\left (x + 4\right )} - 2 \, x + 8\right )}} + \frac {\log \left (x\right )^{2}}{4 \, x}\right )} \] Input:

integrate((((6*x^2+12*x)*exp(4+x)-24*x^2+8*x-16)*log(x)^2+(-12*x*exp(4+x)+ 
16*x^2-8*x+32)*log(x))*exp(-2*log(x)^2/(3*x^2*exp(4+x)-4*x^3+2*x^2-8*x))/( 
9*x^4*exp(4+x)^2+(-24*x^5+12*x^4-48*x^3)*exp(4+x)+16*x^6-16*x^5+68*x^4-32* 
x^3+64*x^2),x, algorithm="maxima")
 

Output:

e^(-x*log(x)^2/(4*x^2 - 3*x*e^(x + 4) - 2*x + 8) + 3/4*e^(x + 4)*log(x)^2/ 
(4*x^2 - 3*x*e^(x + 4) - 2*x + 8) + 1/2*log(x)^2/(4*x^2 - 3*x*e^(x + 4) - 
2*x + 8) + 1/4*log(x)^2/x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=e^{\left (\frac {2 \, \log \left (x\right )^{2}}{4 \, {\left (x + 4\right )}^{3} - 3 \, {\left (x + 4\right )}^{2} e^{\left (x + 4\right )} - 50 \, {\left (x + 4\right )}^{2} + 24 \, {\left (x + 4\right )} e^{\left (x + 4\right )} + 216 \, x - 48 \, e^{\left (x + 4\right )} + 544}\right )} \] Input:

integrate((((6*x^2+12*x)*exp(4+x)-24*x^2+8*x-16)*log(x)^2+(-12*x*exp(4+x)+ 
16*x^2-8*x+32)*log(x))*exp(-2*log(x)^2/(3*x^2*exp(4+x)-4*x^3+2*x^2-8*x))/( 
9*x^4*exp(4+x)^2+(-24*x^5+12*x^4-48*x^3)*exp(4+x)+16*x^6-16*x^5+68*x^4-32* 
x^3+64*x^2),x, algorithm="giac")
 

Output:

e^(2*log(x)^2/(4*(x + 4)^3 - 3*(x + 4)^2*e^(x + 4) - 50*(x + 4)^2 + 24*(x 
+ 4)*e^(x + 4) + 216*x - 48*e^(x + 4) + 544))
 

Mupad [B] (verification not implemented)

Time = 3.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx={\mathrm {e}}^{\frac {2\,{\ln \left (x\right )}^2}{8\,x-2\,x^2+4\,x^3-3\,x^2\,{\mathrm {e}}^4\,{\mathrm {e}}^x}} \] Input:

int((exp((2*log(x)^2)/(8*x - 3*x^2*exp(x + 4) - 2*x^2 + 4*x^3))*(log(x)^2* 
(8*x + exp(x + 4)*(12*x + 6*x^2) - 24*x^2 - 16) - log(x)*(8*x + 12*x*exp(x 
 + 4) - 16*x^2 - 32)))/(9*x^4*exp(2*x + 8) - exp(x + 4)*(48*x^3 - 12*x^4 + 
 24*x^5) + 64*x^2 - 32*x^3 + 68*x^4 - 16*x^5 + 16*x^6),x)
 

Output:

exp((2*log(x)^2)/(8*x - 2*x^2 + 4*x^3 - 3*x^2*exp(4)*exp(x)))
 

Reduce [F]

\[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=\text {too large to display} \] Input:

int((((6*x^2+12*x)*exp(4+x)-24*x^2+8*x-16)*log(x)^2+(-12*x*exp(4+x)+16*x^2 
-8*x+32)*log(x))*exp(-2*log(x)^2/(3*x^2*exp(4+x)-4*x^3+2*x^2-8*x))/(9*x^4* 
exp(4+x)^2+(-24*x^5+12*x^4-48*x^3)*exp(4+x)+16*x^6-16*x^5+68*x^4-32*x^3+64 
*x^2),x)
 

Output:

2*( - 8*int(log(x)**2/(9*e**((6*e**x*e**4*x**3 + 2*log(x)**2 - 8*x**4 + 4* 
x**3 - 16*x**2)/(3*e**x*e**4*x**2 - 4*x**3 + 2*x**2 - 8*x))*e**8*x**4 - 24 
*e**((3*e**x*e**4*x**3 + 2*log(x)**2 - 4*x**4 + 2*x**3 - 8*x**2)/(3*e**x*e 
**4*x**2 - 4*x**3 + 2*x**2 - 8*x))*e**4*x**5 + 12*e**((3*e**x*e**4*x**3 + 
2*log(x)**2 - 4*x**4 + 2*x**3 - 8*x**2)/(3*e**x*e**4*x**2 - 4*x**3 + 2*x** 
2 - 8*x))*e**4*x**4 - 48*e**((3*e**x*e**4*x**3 + 2*log(x)**2 - 4*x**4 + 2* 
x**3 - 8*x**2)/(3*e**x*e**4*x**2 - 4*x**3 + 2*x**2 - 8*x))*e**4*x**3 + 16* 
e**((2*log(x)**2)/(3*e**x*e**4*x**2 - 4*x**3 + 2*x**2 - 8*x))*x**6 - 16*e* 
*((2*log(x)**2)/(3*e**x*e**4*x**2 - 4*x**3 + 2*x**2 - 8*x))*x**5 + 68*e**( 
(2*log(x)**2)/(3*e**x*e**4*x**2 - 4*x**3 + 2*x**2 - 8*x))*x**4 - 32*e**((2 
*log(x)**2)/(3*e**x*e**4*x**2 - 4*x**3 + 2*x**2 - 8*x))*x**3 + 64*e**((2*l 
og(x)**2)/(3*e**x*e**4*x**2 - 4*x**3 + 2*x**2 - 8*x))*x**2),x) + 4*int(log 
(x)**2/(9*e**((6*e**x*e**4*x**3 + 2*log(x)**2 - 8*x**4 + 4*x**3 - 16*x**2) 
/(3*e**x*e**4*x**2 - 4*x**3 + 2*x**2 - 8*x))*e**8*x**3 - 24*e**((3*e**x*e* 
*4*x**3 + 2*log(x)**2 - 4*x**4 + 2*x**3 - 8*x**2)/(3*e**x*e**4*x**2 - 4*x* 
*3 + 2*x**2 - 8*x))*e**4*x**4 + 12*e**((3*e**x*e**4*x**3 + 2*log(x)**2 - 4 
*x**4 + 2*x**3 - 8*x**2)/(3*e**x*e**4*x**2 - 4*x**3 + 2*x**2 - 8*x))*e**4* 
x**3 - 48*e**((3*e**x*e**4*x**3 + 2*log(x)**2 - 4*x**4 + 2*x**3 - 8*x**2)/ 
(3*e**x*e**4*x**2 - 4*x**3 + 2*x**2 - 8*x))*e**4*x**2 + 16*e**((2*log(x)** 
2)/(3*e**x*e**4*x**2 - 4*x**3 + 2*x**2 - 8*x))*x**5 - 16*e**((2*log(x)*...