\(\int \frac {e^{-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} (e^8 x \log ^2(x)+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}} (e^8 (4-x)+e^{8+\frac {x}{e^8}} x+(e^8 x+e^{\frac {x}{e^8}} (-e^8 x-x^2)) \log (x)))}{x \log ^2(x)} \, dx\) [1360]

Optimal result

Integrand size = 114, antiderivative size = 25 \[ \int \frac {e^{-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \left (e^8 x \log ^2(x)+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}} \left (e^8 (4-x)+e^{8+\frac {x}{e^8}} x+\left (e^8 x+e^{\frac {x}{e^8}} \left (-e^8 x-x^2\right )\right ) \log (x)\right )\right )}{x \log ^2(x)} \, dx=e^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \left (e^8 x \log ^2(x)+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}} \left (e^8 (4-x)+e^{8+\frac {x}{e^8}} x+\left (e^8 x+e^{\frac {x}{e^8}} \left (-e^8 x-x^2\right )\right ) \log (x)\right )\right )}{x \log ^2(x)} \, dx=e^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \] Input:

Integrate[(E^(-6 + E^((-4 + x - E^(x/E^8)*x)/Log[x]) + x)*(E^8*x*Log[x]^2 
+ E^((-4 + x - E^(x/E^8)*x)/Log[x])*(E^8*(4 - x) + E^(8 + x/E^8)*x + (E^8* 
x + E^(x/E^8)*(-(E^8*x) - x^2))*Log[x])))/(x*Log[x]^2),x]
 

Output:

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

\(\displaystyle \int \frac {e^{x+e^{\frac {-e^{\frac {x}{e^8}} x+x-4}{\log (x)}}-6} \left (e^{\frac {-e^{\frac {x}{e^8}} x+x-4}{\log (x)}} \left (\left (e^{\frac {x}{e^8}} \left (-x^2-e^8 x\right )+e^8 x\right ) \log (x)+e^8 (4-x)+e^{\frac {x}{e^8}+8} x\right )+e^8 x \log ^2(x)\right )}{x \log ^2(x)} \, dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7299

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

Input:

Int[(E^(-6 + E^((-4 + x - E^(x/E^8)*x)/Log[x]) + x)*(E^8*x*Log[x]^2 + E^(( 
-4 + x - E^(x/E^8)*x)/Log[x])*(E^8*(4 - x) + E^(8 + x/E^8)*x + (E^8*x + E^ 
(x/E^8)*(-(E^8*x) - x^2))*Log[x])))/(x*Log[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 185.77 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{-\frac {x \,{\mathrm e}^{x \,{\mathrm e}^{-8}}-x +4}{\ln \left (x \right )}}+2+x}\)	\(24\)
parallelrisch	\({\mathrm e}^{{\mathrm e}^{-\frac {x \,{\mathrm e}^{x \,{\mathrm e}^{-8}}-x +4}{\ln \left (x \right )}}+2+x}\)	\(26\)

Input:

int(((((-x*exp(4)^2-x^2)*exp(x/exp(4)^2)+x*exp(4)^2)*ln(x)+x*exp(4)^2*exp( 
x/exp(4)^2)+(-x+4)*exp(4)^2)*exp((-x*exp(x/exp(4)^2)+x-4)/ln(x))+x*exp(4)^ 
2*ln(x)^2)*exp(exp((-x*exp(x/exp(4)^2)+x-4)/ln(x))+2+x)/x/exp(4)^2/ln(x)^2 
,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \left (e^8 x \log ^2(x)+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}} \left (e^8 (4-x)+e^{8+\frac {x}{e^8}} x+\left (e^8 x+e^{\frac {x}{e^8}} \left (-e^8 x-x^2\right )\right ) \log (x)\right )\right )}{x \log ^2(x)} \, dx=e^{\left (x + e^{\left (\frac {{\left ({\left (x - 4\right )} e^{8} - x e^{\left ({\left (x + 8 \, e^{8}\right )} e^{\left (-8\right )}\right )}\right )} e^{\left (-8\right )}}{\log \left (x\right )}\right )} + 2\right )} \] Input:

integrate(((((-x*exp(4)^2-x^2)*exp(x/exp(4)^2)+x*exp(4)^2)*log(x)+x*exp(4) 
^2*exp(x/exp(4)^2)+(-x+4)*exp(4)^2)*exp((-x*exp(x/exp(4)^2)+x-4)/log(x))+x 
*exp(4)^2*log(x)^2)*exp(exp((-x*exp(x/exp(4)^2)+x-4)/log(x))+2+x)/x/exp(4) 
^2/log(x)^2,x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 82.88 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {e^{-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \left (e^8 x \log ^2(x)+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}} \left (e^8 (4-x)+e^{8+\frac {x}{e^8}} x+\left (e^8 x+e^{\frac {x}{e^8}} \left (-e^8 x-x^2\right )\right ) \log (x)\right )\right )}{x \log ^2(x)} \, dx=e^{x + e^{\frac {- x e^{\frac {x}{e^{8}}} + x - 4}{\log {\left (x \right )}}} + 2} \] Input:

integrate(((((-x*exp(4)**2-x**2)*exp(x/exp(4)**2)+x*exp(4)**2)*ln(x)+x*exp 
(4)**2*exp(x/exp(4)**2)+(-x+4)*exp(4)**2)*exp((-x*exp(x/exp(4)**2)+x-4)/ln 
(x))+x*exp(4)**2*ln(x)**2)*exp(exp((-x*exp(x/exp(4)**2)+x-4)/ln(x))+2+x)/x 
/exp(4)**2/ln(x)**2,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {e^{-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \left (e^8 x \log ^2(x)+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}} \left (e^8 (4-x)+e^{8+\frac {x}{e^8}} x+\left (e^8 x+e^{\frac {x}{e^8}} \left (-e^8 x-x^2\right )\right ) \log (x)\right )\right )}{x \log ^2(x)} \, dx=e^{\left (x + e^{\left (-\frac {x e^{\left (x e^{\left (-8\right )}\right )}}{\log \left (x\right )} + \frac {x}{\log \left (x\right )} - \frac {4}{\log \left (x\right )}\right )} + 2\right )} \] Input:

integrate(((((-x*exp(4)^2-x^2)*exp(x/exp(4)^2)+x*exp(4)^2)*log(x)+x*exp(4) 
^2*exp(x/exp(4)^2)+(-x+4)*exp(4)^2)*exp((-x*exp(x/exp(4)^2)+x-4)/log(x))+x 
*exp(4)^2*log(x)^2)*exp(exp((-x*exp(x/exp(4)^2)+x-4)/log(x))+2+x)/x/exp(4) 
^2/log(x)^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {e^{-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \left (e^8 x \log ^2(x)+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}} \left (e^8 (4-x)+e^{8+\frac {x}{e^8}} x+\left (e^8 x+e^{\frac {x}{e^8}} \left (-e^8 x-x^2\right )\right ) \log (x)\right )\right )}{x \log ^2(x)} \, dx=\int { \frac {{\left (x e^{8} \log \left (x\right )^{2} - {\left ({\left (x - 4\right )} e^{8} - x e^{\left (x e^{\left (-8\right )} + 8\right )} - {\left (x e^{8} - {\left (x^{2} + x e^{8}\right )} e^{\left (x e^{\left (-8\right )}\right )}\right )} \log \left (x\right )\right )} e^{\left (-\frac {x e^{\left (x e^{\left (-8\right )}\right )} - x + 4}{\log \left (x\right )}\right )}\right )} e^{\left (x + e^{\left (-\frac {x e^{\left (x e^{\left (-8\right )}\right )} - x + 4}{\log \left (x\right )}\right )} - 6\right )}}{x \log \left (x\right )^{2}} \,d x } \] Input:

integrate(((((-x*exp(4)^2-x^2)*exp(x/exp(4)^2)+x*exp(4)^2)*log(x)+x*exp(4) 
^2*exp(x/exp(4)^2)+(-x+4)*exp(4)^2)*exp((-x*exp(x/exp(4)^2)+x-4)/log(x))+x 
*exp(4)^2*log(x)^2)*exp(exp((-x*exp(x/exp(4)^2)+x-4)/log(x))+2+x)/x/exp(4) 
^2/log(x)^2,x, algorithm="giac")
 

Output:

undef
 

Mupad [B] (verification not implemented)

Time = 3.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \left (e^8 x \log ^2(x)+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}} \left (e^8 (4-x)+e^{8+\frac {x}{e^8}} x+\left (e^8 x+e^{\frac {x}{e^8}} \left (-e^8 x-x^2\right )\right ) \log (x)\right )\right )}{x \log ^2(x)} \, dx={\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^{-\frac {4}{\ln \left (x\right )}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-8}}}{\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {x}{\ln \left (x\right )}}}\,{\mathrm {e}}^x \] Input:

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

Output:

exp(2)*exp(exp(-4/log(x))*exp(-(x*exp(x*exp(-8)))/log(x))*exp(x/log(x)))*e 
xp(x)
 

Reduce [F]

\[ \int \frac {e^{-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \left (e^8 x \log ^2(x)+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}} \left (e^8 (4-x)+e^{8+\frac {x}{e^8}} x+\left (e^8 x+e^{\frac {x}{e^8}} \left (-e^8 x-x^2\right )\right ) \log (x)\right )\right )}{x \log ^2(x)} \, dx =\text {Too large to display} \] Input:

int(((((-x*exp(4)^2-x^2)*exp(x/exp(4)^2)+x*exp(4)^2)*log(x)+x*exp(4)^2*exp 
(x/exp(4)^2)+(-x+4)*exp(4)^2)*exp((-x*exp(x/exp(4)^2)+x-4)/log(x))+x*exp(4 
)^2*log(x)^2)*exp(exp((-x*exp(x/exp(4)^2)+x-4)/log(x))+2+x)/x/exp(4)^2/log 
(x)^2,x)
 

Output:

(int(e**((e**((e**(x/e**8)*x + 4)/log(x))*x + e**(x/log(x)))/e**((e**(x/e* 
*8)*x + 4)/log(x))),x)*e**8 + int(e**((e**((e**(x/e**8)*x + 4)/log(x))*log 
(x)*e**8*x + e**((e**(x/e**8)*x + 4)/log(x))*log(x)*x + e**((e**(x/e**8)*x 
 + 4)/log(x))*e**8*x + e**(x/log(x))*log(x)*e**8)/(e**((e**(x/e**8)*x + 4) 
/log(x))*log(x)*e**8))/(e**((e**(x/e**8)*x + 4)/log(x))*log(x)**2),x)*e**8 
 - int(e**((e**((e**(x/e**8)*x + 4)/log(x))*log(x)*e**8*x + e**((e**(x/e** 
8)*x + 4)/log(x))*log(x)*x + e**((e**(x/e**8)*x + 4)/log(x))*e**8*x + e**( 
x/log(x))*log(x)*e**8)/(e**((e**(x/e**8)*x + 4)/log(x))*log(x)*e**8))/(e** 
((e**(x/e**8)*x + 4)/log(x))*log(x)),x)*e**8 + 4*int(e**((e**((e**(x/e**8) 
*x + 4)/log(x))*log(x)*x + e**((e**(x/e**8)*x + 4)/log(x))*x + e**(x/log(x 
))*log(x))/(e**((e**(x/e**8)*x + 4)/log(x))*log(x)))/(e**((e**(x/e**8)*x + 
 4)/log(x))*log(x)**2*x),x)*e**8 - int(e**((e**((e**(x/e**8)*x + 4)/log(x) 
)*log(x)*x + e**((e**(x/e**8)*x + 4)/log(x))*x + e**(x/log(x))*log(x))/(e* 
*((e**(x/e**8)*x + 4)/log(x))*log(x)))/(e**((e**(x/e**8)*x + 4)/log(x))*lo 
g(x)**2),x)*e**8 + int(e**((e**((e**(x/e**8)*x + 4)/log(x))*log(x)*x + e** 
((e**(x/e**8)*x + 4)/log(x))*x + e**(x/log(x))*log(x))/(e**((e**(x/e**8)*x 
 + 4)/log(x))*log(x)))/(e**((e**(x/e**8)*x + 4)/log(x))*log(x)),x)*e**8 - 
int((e**((e**((e**(x/e**8)*x + 4)/log(x))*log(x)*e**8*x + e**((e**(x/e**8) 
*x + 4)/log(x))*log(x)*x + e**((e**(x/e**8)*x + 4)/log(x))*e**8*x + e**(x/ 
log(x))*log(x)*e**8)/(e**((e**(x/e**8)*x + 4)/log(x))*log(x)*e**8))*x)/...