\(\int \frac {e^{\frac {-7 x+e^{36+e^4+12 x^2+x^4+e^2 (-12-2 x^2)} x}{16+\log (x)}} (-105+e^{36+e^4+12 x^2+x^4+e^2 (-12-2 x^2)} (15+384 x^2-64 e^2 x^2+64 x^4)+(-7+e^{36+e^4+12 x^2+x^4+e^2 (-12-2 x^2)} (1+24 x^2-4 e^2 x^2+4 x^4)) \log (x))}{256+32 \log (x)+\log ^2(x)} \, dx\) [2649]

Optimal result

Integrand size = 155, antiderivative size = 26 \[ \int \frac {e^{\frac {-7 x+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} x}{16+\log (x)}} \left (-105+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (15+384 x^2-64 e^2 x^2+64 x^4\right )+\left (-7+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (1+24 x^2-4 e^2 x^2+4 x^4\right )\right ) \log (x)\right )}{256+32 \log (x)+\log ^2(x)} \, dx=e^{\frac {\left (-7+e^{\left (-6+e^2-x^2\right )^2}\right ) x}{16+\log (x)}} \] Output:

exp(x/(16+ln(x))*(exp((exp(2)-x^2-6)^2)-7))
 

Mathematica [F]

\[ \int \frac {e^{\frac {-7 x+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} x}{16+\log (x)}} \left (-105+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (15+384 x^2-64 e^2 x^2+64 x^4\right )+\left (-7+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (1+24 x^2-4 e^2 x^2+4 x^4\right )\right ) \log (x)\right )}{256+32 \log (x)+\log ^2(x)} \, dx=\int \frac {e^{\frac {-7 x+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} x}{16+\log (x)}} \left (-105+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (15+384 x^2-64 e^2 x^2+64 x^4\right )+\left (-7+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (1+24 x^2-4 e^2 x^2+4 x^4\right )\right ) \log (x)\right )}{256+32 \log (x)+\log ^2(x)} \, dx \] Input:

Integrate[(E^((-7*x + E^(36 + E^4 + 12*x^2 + x^4 + E^2*(-12 - 2*x^2))*x)/( 
16 + Log[x]))*(-105 + E^(36 + E^4 + 12*x^2 + x^4 + E^2*(-12 - 2*x^2))*(15 
+ 384*x^2 - 64*E^2*x^2 + 64*x^4) + (-7 + E^(36 + E^4 + 12*x^2 + x^4 + E^2* 
(-12 - 2*x^2))*(1 + 24*x^2 - 4*E^2*x^2 + 4*x^4))*Log[x]))/(256 + 32*Log[x] 
 + Log[x]^2),x]
 

Output:

Integrate[(E^((-7*x + E^(36 + E^4 + 12*x^2 + x^4 + E^2*(-12 - 2*x^2))*x)/( 
16 + Log[x]))*(-105 + E^(36 + E^4 + 12*x^2 + x^4 + E^2*(-12 - 2*x^2))*(15 
+ 384*x^2 - 64*E^2*x^2 + 64*x^4) + (-7 + E^(36 + E^4 + 12*x^2 + x^4 + E^2* 
(-12 - 2*x^2))*(1 + 24*x^2 - 4*E^2*x^2 + 4*x^4))*Log[x]))/(256 + 32*Log[x] 
 + Log[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.

\(\displaystyle \int \frac {\left (e^{x^4+12 x^2+e^2 \left (-2 x^2-12\right )+e^4+36} \left (64 x^4-64 e^2 x^2+384 x^2+15\right )+\left (e^{x^4+12 x^2+e^2 \left (-2 x^2-12\right )+e^4+36} \left (4 x^4-4 e^2 x^2+24 x^2+1\right )-7\right ) \log (x)-105\right ) \exp \left (\frac {e^{x^4+12 x^2+e^2 \left (-2 x^2-12\right )+e^4+36} x-7 x}{\log (x)+16}\right )}{\log ^2(x)+32 \log (x)+256} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (e^{x^4+12 x^2+e^2 \left (-2 x^2-12\right )+e^4+36} \left (64 x^4-64 e^2 x^2+384 x^2+15\right )+\left (e^{x^4+12 x^2+e^2 \left (-2 x^2-12\right )+e^4+36} \left (4 x^4-4 e^2 x^2+24 x^2+1\right )-7\right ) \log (x)-105\right ) \exp \left (\frac {e^{x^4+12 x^2+e^2 \left (-2 x^2-12\right )+e^4+36} x-7 x}{\log (x)+16}\right )}{(\log (x)+16)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (64 x^4+4 x^4 \log (x)+384 \left (1-\frac {e^2}{6}\right ) x^2+24 \left (1-\frac {e^2}{6}\right ) x^2 \log (x)+\log (x)+15\right ) \exp \left (x^4+2 \left (6-e^2\right ) x^2+\frac {e^{x^4+12 x^2+e^2 \left (-2 x^2-12\right )+e^4+36} x-7 x}{\log (x)+16}+\left (e^2-6\right )^2\right )}{(\log (x)+16)^2}-\frac {7 (\log (x)+15) \exp \left (\frac {e^{x^4+12 x^2+e^2 \left (-2 x^2-12\right )+e^4+36} x-7 x}{\log (x)+16}\right )}{(\log (x)+16)^2}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (64 x^4+4 x^4 \log (x)+384 \left (1-\frac {e^2}{6}\right ) x^2+24 \left (1-\frac {e^2}{6}\right ) x^2 \log (x)+\log (x)+15\right ) \exp \left (x^4-2 \left (e^2-6\right ) x^2+\frac {e^{\left (x^2-e^2+6\right )^2} x}{\log (x)+16}-\frac {7 x}{\log (x)+16}+\left (e^2-6\right )^2\right )}{(\log (x)+16)^2}-\frac {7 e^{\frac {\left (e^{\left (x^2-e^2+6\right )^2}-7\right ) x}{\log (x)+16}} (\log (x)+15)}{(\log (x)+16)^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {\left (64 x^4+4 x^4 \log (x)+384 \left (1-\frac {e^2}{6}\right ) x^2+24 \left (1-\frac {e^2}{6}\right ) x^2 \log (x)+\log (x)+15\right ) \exp \left (x^4-2 \left (e^2-6\right ) x^2+\frac {e^{\left (x^2-e^2+6\right )^2} x}{\log (x)+16}-\frac {7 x}{\log (x)+16}+\left (e^2-6\right )^2\right )}{(\log (x)+16)^2}-\frac {7 e^{\frac {\left (e^{\left (x^2-e^2+6\right )^2}-7\right ) x}{\log (x)+16}} (\log (x)+15)}{(\log (x)+16)^2}\right )dx\)

Input:

Int[(E^((-7*x + E^(36 + E^4 + 12*x^2 + x^4 + E^2*(-12 - 2*x^2))*x)/(16 + L 
og[x]))*(-105 + E^(36 + E^4 + 12*x^2 + x^4 + E^2*(-12 - 2*x^2))*(15 + 384* 
x^2 - 64*E^2*x^2 + 64*x^4) + (-7 + E^(36 + E^4 + 12*x^2 + x^4 + E^2*(-12 - 
 2*x^2))*(1 + 24*x^2 - 4*E^2*x^2 + 4*x^4))*Log[x]))/(256 + 32*Log[x] + Log 
[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 108.62 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38

method	result	size
risch	\({\mathrm e}^{\frac {x \left ({\mathrm e}^{x^{4}-2 x^{2} {\mathrm e}^{2}+12 x^{2}+{\mathrm e}^{4}-12 \,{\mathrm e}^{2}+36}-7\right )}{16+\ln \left (x \right )}}\)	\(36\)
parallelrisch	\({\mathrm e}^{\frac {x \left ({\mathrm e}^{x^{4}-2 x^{2} {\mathrm e}^{2}+12 x^{2}+{\mathrm e}^{4}-12 \,{\mathrm e}^{2}+36}-7\right )}{16+\ln \left (x \right )}}\)	\(38\)

Input:

int((((-4*x^2*exp(2)+4*x^4+24*x^2+1)*exp(exp(2)^2+(-2*x^2-12)*exp(2)+x^4+1 
2*x^2+36)-7)*ln(x)+(-64*x^2*exp(2)+64*x^4+384*x^2+15)*exp(exp(2)^2+(-2*x^2 
-12)*exp(2)+x^4+12*x^2+36)-105)*exp((x*exp(exp(2)^2+(-2*x^2-12)*exp(2)+x^4 
+12*x^2+36)-7*x)/(16+ln(x)))/(ln(x)^2+32*ln(x)+256),x,method=_RETURNVERBOS 
E)
 

Output:

exp(x*(exp(x^4-2*x^2*exp(2)+12*x^2+exp(4)-12*exp(2)+36)-7)/(16+ln(x)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {e^{\frac {-7 x+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} x}{16+\log (x)}} \left (-105+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (15+384 x^2-64 e^2 x^2+64 x^4\right )+\left (-7+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (1+24 x^2-4 e^2 x^2+4 x^4\right )\right ) \log (x)\right )}{256+32 \log (x)+\log ^2(x)} \, dx=e^{\left (\frac {x e^{\left (x^{4} + 12 \, x^{2} - 2 \, {\left (x^{2} + 6\right )} e^{2} + e^{4} + 36\right )} - 7 \, x}{\log \left (x\right ) + 16}\right )} \] Input:

integrate((((-4*x^2*exp(2)+4*x^4+24*x^2+1)*exp(exp(2)^2+(-2*x^2-12)*exp(2) 
+x^4+12*x^2+36)-7)*log(x)+(-64*x^2*exp(2)+64*x^4+384*x^2+15)*exp(exp(2)^2+ 
(-2*x^2-12)*exp(2)+x^4+12*x^2+36)-105)*exp((x*exp(exp(2)^2+(-2*x^2-12)*exp 
(2)+x^4+12*x^2+36)-7*x)/(16+log(x)))/(log(x)^2+32*log(x)+256),x, algorithm 
="fricas")
 

Output:

e^((x*e^(x^4 + 12*x^2 - 2*(x^2 + 6)*e^2 + e^4 + 36) - 7*x)/(log(x) + 16))
 

Sympy [A] (verification not implemented)

Time = 1.57 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {e^{\frac {-7 x+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} x}{16+\log (x)}} \left (-105+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (15+384 x^2-64 e^2 x^2+64 x^4\right )+\left (-7+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (1+24 x^2-4 e^2 x^2+4 x^4\right )\right ) \log (x)\right )}{256+32 \log (x)+\log ^2(x)} \, dx=e^{\frac {x e^{x^{4} + 12 x^{2} + \left (- 2 x^{2} - 12\right ) e^{2} + 36 + e^{4}} - 7 x}{\log {\left (x \right )} + 16}} \] Input:

integrate((((-4*x**2*exp(2)+4*x**4+24*x**2+1)*exp(exp(2)**2+(-2*x**2-12)*e 
xp(2)+x**4+12*x**2+36)-7)*ln(x)+(-64*x**2*exp(2)+64*x**4+384*x**2+15)*exp( 
exp(2)**2+(-2*x**2-12)*exp(2)+x**4+12*x**2+36)-105)*exp((x*exp(exp(2)**2+( 
-2*x**2-12)*exp(2)+x**4+12*x**2+36)-7*x)/(16+ln(x)))/(ln(x)**2+32*ln(x)+25 
6),x)
 

Output:

exp((x*exp(x**4 + 12*x**2 + (-2*x**2 - 12)*exp(2) + 36 + exp(4)) - 7*x)/(l 
og(x) + 16))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {-7 x+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} x}{16+\log (x)}} \left (-105+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (15+384 x^2-64 e^2 x^2+64 x^4\right )+\left (-7+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (1+24 x^2-4 e^2 x^2+4 x^4\right )\right ) \log (x)\right )}{256+32 \log (x)+\log ^2(x)} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((((-4*x^2*exp(2)+4*x^4+24*x^2+1)*exp(exp(2)^2+(-2*x^2-12)*exp(2) 
+x^4+12*x^2+36)-7)*log(x)+(-64*x^2*exp(2)+64*x^4+384*x^2+15)*exp(exp(2)^2+ 
(-2*x^2-12)*exp(2)+x^4+12*x^2+36)-105)*exp((x*exp(exp(2)^2+(-2*x^2-12)*exp 
(2)+x^4+12*x^2+36)-7*x)/(16+log(x)))/(log(x)^2+32*log(x)+256),x, algorithm 
="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: In function CAR, the value of 
the first argument is  0which is not of the expected type LIST
 

Giac [F]

\[ \int \frac {e^{\frac {-7 x+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} x}{16+\log (x)}} \left (-105+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (15+384 x^2-64 e^2 x^2+64 x^4\right )+\left (-7+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (1+24 x^2-4 e^2 x^2+4 x^4\right )\right ) \log (x)\right )}{256+32 \log (x)+\log ^2(x)} \, dx=\int { \frac {{\left ({\left (64 \, x^{4} - 64 \, x^{2} e^{2} + 384 \, x^{2} + 15\right )} e^{\left (x^{4} + 12 \, x^{2} - 2 \, {\left (x^{2} + 6\right )} e^{2} + e^{4} + 36\right )} + {\left ({\left (4 \, x^{4} - 4 \, x^{2} e^{2} + 24 \, x^{2} + 1\right )} e^{\left (x^{4} + 12 \, x^{2} - 2 \, {\left (x^{2} + 6\right )} e^{2} + e^{4} + 36\right )} - 7\right )} \log \left (x\right ) - 105\right )} e^{\left (\frac {x e^{\left (x^{4} + 12 \, x^{2} - 2 \, {\left (x^{2} + 6\right )} e^{2} + e^{4} + 36\right )} - 7 \, x}{\log \left (x\right ) + 16}\right )}}{\log \left (x\right )^{2} + 32 \, \log \left (x\right ) + 256} \,d x } \] Input:

integrate((((-4*x^2*exp(2)+4*x^4+24*x^2+1)*exp(exp(2)^2+(-2*x^2-12)*exp(2) 
+x^4+12*x^2+36)-7)*log(x)+(-64*x^2*exp(2)+64*x^4+384*x^2+15)*exp(exp(2)^2+ 
(-2*x^2-12)*exp(2)+x^4+12*x^2+36)-105)*exp((x*exp(exp(2)^2+(-2*x^2-12)*exp 
(2)+x^4+12*x^2+36)-7*x)/(16+log(x)))/(log(x)^2+32*log(x)+256),x, algorithm 
="giac")
 

Output:

integrate(((64*x^4 - 64*x^2*e^2 + 384*x^2 + 15)*e^(x^4 + 12*x^2 - 2*(x^2 + 
 6)*e^2 + e^4 + 36) + ((4*x^4 - 4*x^2*e^2 + 24*x^2 + 1)*e^(x^4 + 12*x^2 - 
2*(x^2 + 6)*e^2 + e^4 + 36) - 7)*log(x) - 105)*e^((x*e^(x^4 + 12*x^2 - 2*( 
x^2 + 6)*e^2 + e^4 + 36) - 7*x)/(log(x) + 16))/(log(x)^2 + 32*log(x) + 256 
), x)
 

Mupad [B] (verification not implemented)

Time = 3.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {e^{\frac {-7 x+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} x}{16+\log (x)}} \left (-105+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (15+384 x^2-64 e^2 x^2+64 x^4\right )+\left (-7+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (1+24 x^2-4 e^2 x^2+4 x^4\right )\right ) \log (x)\right )}{256+32 \log (x)+\log ^2(x)} \, dx={\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^2}\,{\mathrm {e}}^{-12\,{\mathrm {e}}^2}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{36}\,{\mathrm {e}}^{12\,x^2}\,{\mathrm {e}}^{{\mathrm {e}}^4}}{\ln \left (x\right )+16}}\,{\mathrm {e}}^{-\frac {7\,x}{\ln \left (x\right )+16}} \] Input:

int((exp(-(7*x - x*exp(exp(4) - exp(2)*(2*x^2 + 12) + 12*x^2 + x^4 + 36))/ 
(log(x) + 16))*(log(x)*(exp(exp(4) - exp(2)*(2*x^2 + 12) + 12*x^2 + x^4 + 
36)*(24*x^2 - 4*x^2*exp(2) + 4*x^4 + 1) - 7) + exp(exp(4) - exp(2)*(2*x^2 
+ 12) + 12*x^2 + x^4 + 36)*(384*x^2 - 64*x^2*exp(2) + 64*x^4 + 15) - 105)) 
/(32*log(x) + log(x)^2 + 256),x)
 

Output:

exp((x*exp(-2*x^2*exp(2))*exp(-12*exp(2))*exp(x^4)*exp(36)*exp(12*x^2)*exp 
(exp(4)))/(log(x) + 16))*exp(-(7*x)/(log(x) + 16))
 

Reduce [F]

\[ \int \frac {e^{\frac {-7 x+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} x}{16+\log (x)}} \left (-105+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (15+384 x^2-64 e^2 x^2+64 x^4\right )+\left (-7+e^{36+e^4+12 x^2+x^4+e^2 \left (-12-2 x^2\right )} \left (1+24 x^2-4 e^2 x^2+4 x^4\right )\right ) \log (x)\right )}{256+32 \log (x)+\log ^2(x)} \, dx=\text {too large to display} \] Input:

int((((-4*x^2*exp(2)+4*x^4+24*x^2+1)*exp(exp(2)^2+(-2*x^2-12)*exp(2)+x^4+1 
2*x^2+36)-7)*log(x)+(-64*x^2*exp(2)+64*x^4+384*x^2+15)*exp(exp(2)^2+(-2*x^ 
2-12)*exp(2)+x^4+12*x^2+36)-105)*exp((x*exp(exp(2)^2+(-2*x^2-12)*exp(2)+x^ 
4+12*x^2+36)-7*x)/(16+log(x)))/(log(x)^2+32*log(x)+256),x)
 

Output:

15*e**(e**4)*int(e**((e**(e**4 + x**4 + 12*x**2)*e**36*x + e**(2*e**2*x**2 
 + 12*e**2)*log(x)*x**4 + 12*e**(2*e**2*x**2 + 12*e**2)*log(x)*x**2 + 16*e 
**(2*e**2*x**2 + 12*e**2)*x**4 + 192*e**(2*e**2*x**2 + 12*e**2)*x**2)/(e** 
(2*e**2*x**2 + 12*e**2)*log(x) + 16*e**(2*e**2*x**2 + 12*e**2)))/(e**((2*l 
og(x)*e**2*x**2 + 12*log(x)*e**2 + 32*e**2*x**2 + 192*e**2 + 7*x)/(log(x) 
+ 16))*log(x)**2 + 32*e**((2*log(x)*e**2*x**2 + 12*log(x)*e**2 + 32*e**2*x 
**2 + 192*e**2 + 7*x)/(log(x) + 16))*log(x) + 256*e**((2*log(x)*e**2*x**2 
+ 12*log(x)*e**2 + 32*e**2*x**2 + 192*e**2 + 7*x)/(log(x) + 16))),x)*e**36 
 + 64*e**(e**4)*int((e**((e**(e**4 + x**4 + 12*x**2)*e**36*x + e**(2*e**2* 
x**2 + 12*e**2)*log(x)*x**4 + 12*e**(2*e**2*x**2 + 12*e**2)*log(x)*x**2 + 
16*e**(2*e**2*x**2 + 12*e**2)*x**4 + 192*e**(2*e**2*x**2 + 12*e**2)*x**2)/ 
(e**(2*e**2*x**2 + 12*e**2)*log(x) + 16*e**(2*e**2*x**2 + 12*e**2)))*x**4) 
/(e**((2*log(x)*e**2*x**2 + 12*log(x)*e**2 + 32*e**2*x**2 + 192*e**2 + 7*x 
)/(log(x) + 16))*log(x)**2 + 32*e**((2*log(x)*e**2*x**2 + 12*log(x)*e**2 + 
 32*e**2*x**2 + 192*e**2 + 7*x)/(log(x) + 16))*log(x) + 256*e**((2*log(x)* 
e**2*x**2 + 12*log(x)*e**2 + 32*e**2*x**2 + 192*e**2 + 7*x)/(log(x) + 16)) 
),x)*e**36 - 64*e**(e**4)*int((e**((e**(e**4 + x**4 + 12*x**2)*e**36*x + e 
**(2*e**2*x**2 + 12*e**2)*log(x)*x**4 + 12*e**(2*e**2*x**2 + 12*e**2)*log( 
x)*x**2 + 16*e**(2*e**2*x**2 + 12*e**2)*x**4 + 192*e**(2*e**2*x**2 + 12*e* 
*2)*x**2)/(e**(2*e**2*x**2 + 12*e**2)*log(x) + 16*e**(2*e**2*x**2 + 12*...