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

3.1.28.1 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)
 

3.1.28.2 Mathematica [A] (verified)

Time = 0.17 (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)))
 

3.1.28.3 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
 

3.1.28.3.1 Defintions of rubi rules used

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.1.28.4 Maple [A] (verified)

Time = 133.61 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97

method	result	size
risch	\({\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{x \left (-3 x \,{\mathrm e}^{4+x}+4 x^{2}-2 x +8\right )}}\)	\(30\)
parallelrisch	\({\mathrm e}^{-\frac {2 \ln \left (x \right )^{2}}{x \left (3 x \,{\mathrm e}^{4+x}-4 x^{2}+2 x -8\right )}}\)	\(30\)

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,method=_RETURNVERBOSE)
 

output

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

3.1.28.5 Fricas [A] (verification not implemented)

Time = 0.25 (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=\
 

output

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

3.1.28.6 Sympy [A] (verification not implemented)

Time = 1.47 (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))
 

3.1.28.7 Maxima [B] (verification not implemented)

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

Time = 0.50 (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=\
 

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)
 

3.1.28.8 Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=\text {Exception raised: RuntimeError} \]

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=\
 

output

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{1741 
4258688,[0,8,9,38,8]%%%}+%%%{-156728328192,[0,8,9,37,8]%%%}+%%%{7967023349 
76,[0,8,9,36
 

3.1.28.9 Mupad [B] (verification not implemented)

Time = 12.36 (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)))