\(\int \frac {-e^{e^x} x+e^{e^{-e^x} (x+e^{e^x} (1+x))} (-x-e^{e^x} x+e^x x^2)+(e^{e^x+e^{-e^x} (x+e^{e^x} (1+x))}+e^{e^x} x) \log (e^{e^{-e^x} (x+e^{e^x} (1+x))}+x)}{(e^{e^x+e^{-e^x} (x+e^{e^x} (1+x))}+e^{e^x} x) \log ^2(e^{e^{-e^x} (x+e^{e^x} (1+x))}+x)} \, dx\) [267]

Optimal result

Integrand size = 170, antiderivative size = 21 \[ \int \frac {-e^{e^x} x+e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )} \left (-x-e^{e^x} x+e^x x^2\right )+\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log \left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )}{\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log ^2\left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )} \, dx=\frac {x}{\log \left (e^{1+x+e^{-e^x} x}+x\right )} \] Output:

x/ln(x+exp(x/exp(exp(x))+1+x))
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-e^{e^x} x+e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )} \left (-x-e^{e^x} x+e^x x^2\right )+\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log \left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )}{\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log ^2\left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )} \, dx=\frac {x}{\log \left (e^{1+x+e^{-e^x} x}+x\right )} \] Input:

Integrate[(-(E^E^x*x) + E^((x + E^E^x*(1 + x))/E^E^x)*(-x - E^E^x*x + E^x* 
x^2) + (E^(E^x + (x + E^E^x*(1 + x))/E^E^x) + E^E^x*x)*Log[E^((x + E^E^x*( 
1 + x))/E^E^x) + x])/((E^(E^x + (x + E^E^x*(1 + x))/E^E^x) + E^E^x*x)*Log[ 
E^((x + E^E^x*(1 + x))/E^E^x) + x]^2),x]
 

Output:

x/Log[E^(1 + x + x/E^E^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.

\(\displaystyle \int \frac {e^{e^{-e^x} \left (x+e^{e^x} (x+1)\right )} \left (e^x x^2-e^{e^x} x-x\right )-e^{e^x} x+\left (e^{e^x} x+e^{e^{-e^x} \left (x+e^{e^x} (x+1)\right )+e^x}\right ) \log \left (x+e^{e^{-e^x} \left (x+e^{e^x} (x+1)\right )}\right )}{\left (e^{e^x} x+e^{e^{-e^x} \left (x+e^{e^x} (x+1)\right )+e^x}\right ) \log ^2\left (x+e^{e^{-e^x} \left (x+e^{e^x} (x+1)\right )}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{-e^x} \left (e^{e^{-e^x} \left (x+e^{e^x} (x+1)\right )} \left (e^x x^2-e^{e^x} x-x\right )-e^{e^x} x+\left (e^{e^x} x+e^{e^{-e^x} \left (x+e^{e^x} (x+1)\right )+e^x}\right ) \log \left (x+e^{e^{-e^x} \left (x+e^{e^x} (x+1)\right )}\right )\right )}{\left (x+e^{e^{-e^x} x+x+1}\right ) \log ^2\left (x+e^{e^{-e^x} \left (x+e^{e^x} (x+1)\right )}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{-e^x} \left (e^x x^2-e^{e^x} x-x+e^{e^x} \log \left (x+e^{e^{-e^x} x+x+1}\right )\right )}{\log ^2\left (x+e^{e^{-e^x} x+x+1}\right )}-\frac {e^{-e^x} x \left (e^x x^2-e^{e^x} x-x+e^{e^x}\right )}{\left (x+e^{e^{-e^x} x+x+1}\right ) \log ^2\left (x+e^{e^{-e^x} x+x+1}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\int \frac {e^{x-e^x} x^3}{\left (x+e^{e^{-e^x} x+x+1}\right ) \log ^2\left (x+e^{e^{-e^x} x+x+1}\right )}dx+\int \frac {e^{x-e^x} x^2}{\log ^2\left (x+e^{e^{-e^x} x+x+1}\right )}dx+\int \frac {x^2}{\left (x+e^{e^{-e^x} x+x+1}\right ) \log ^2\left (x+e^{e^{-e^x} x+x+1}\right )}dx+\int \frac {e^{-e^x} x^2}{\left (x+e^{e^{-e^x} x+x+1}\right ) \log ^2\left (x+e^{e^{-e^x} x+x+1}\right )}dx-\int \frac {x}{\log ^2\left (x+e^{e^{-e^x} x+x+1}\right )}dx-\int \frac {e^{-e^x} x}{\log ^2\left (x+e^{e^{-e^x} x+x+1}\right )}dx-\int \frac {x}{\left (x+e^{e^{-e^x} x+x+1}\right ) \log ^2\left (x+e^{e^{-e^x} x+x+1}\right )}dx+\int \frac {1}{\log \left (x+e^{e^{-e^x} x+x+1}\right )}dx\)

Input:

Int[(-(E^E^x*x) + E^((x + E^E^x*(1 + x))/E^E^x)*(-x - E^E^x*x + E^x*x^2) + 
 (E^(E^x + (x + E^E^x*(1 + x))/E^E^x) + E^E^x*x)*Log[E^((x + E^E^x*(1 + x) 
)/E^E^x) + x])/((E^(E^x + (x + E^E^x*(1 + x))/E^E^x) + E^E^x*x)*Log[E^((x 
+ E^E^x*(1 + x))/E^E^x) + x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.83 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14

method	result	size
parallelrisch	\(\frac {x}{\ln \left ({\mathrm e}^{\left (\left (1+x \right ) {\mathrm e}^{{\mathrm e}^{x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{x}}}+x \right )}\)	\(24\)
risch	\(\frac {x}{\ln \left ({\mathrm e}^{\left (x \,{\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{{\mathrm e}^{x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{x}}}+x \right )}\)	\(25\)

Input:

int(((exp(exp(x))*exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))*ln 
(exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x)+(-x*exp(exp(x))+exp(x)*x^2-x)*e 
xp(((1+x)*exp(exp(x))+x)/exp(exp(x)))-x*exp(exp(x)))/(exp(exp(x))*exp(((1+ 
x)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))/ln(exp(((1+x)*exp(exp(x))+x) 
/exp(exp(x)))+x)^2,x,method=_RETURNVERBOSE)
 

Output:

x/ln(exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \[ \int \frac {-e^{e^x} x+e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )} \left (-x-e^{e^x} x+e^x x^2\right )+\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log \left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )}{\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log ^2\left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )} \, dx=\frac {x}{\log \left ({\left (x e^{\left (e^{x}\right )} + e^{\left ({\left ({\left (x + e^{x} + 1\right )} e^{\left (e^{x}\right )} + x\right )} e^{\left (-e^{x}\right )}\right )}\right )} e^{\left (-e^{x}\right )}\right )} \] Input:

integrate(((exp(exp(x))*exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x 
)))*log(exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x)+(-x*exp(exp(x))+exp(x)*x 
^2-x)*exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))-x*exp(exp(x)))/(exp(exp(x))*e 
xp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))/log(exp(((1+x)*exp(ex 
p(x))+x)/exp(exp(x)))+x)^2,x, algorithm="fricas")
 

Output:

x/log((x*e^(e^x) + e^(((x + e^x + 1)*e^(e^x) + x)*e^(-e^x)))*e^(-e^x))
 

Sympy [A] (verification not implemented)

Time = 4.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-e^{e^x} x+e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )} \left (-x-e^{e^x} x+e^x x^2\right )+\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log \left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )}{\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log ^2\left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )} \, dx=\frac {x}{\log {\left (x + e^{\left (x + \left (x + 1\right ) e^{e^{x}}\right ) e^{- e^{x}}} \right )}} \] Input:

integrate(((exp(exp(x))*exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x 
)))*ln(exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x)+(-x*exp(exp(x))+exp(x)*x* 
*2-x)*exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))-x*exp(exp(x)))/(exp(exp(x))*e 
xp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))/ln(exp(((1+x)*exp(exp 
(x))+x)/exp(exp(x)))+x)**2,x)
 

Output:

x/log(x + exp((x + (x + 1)*exp(exp(x)))*exp(-exp(x))))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {-e^{e^x} x+e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )} \left (-x-e^{e^x} x+e^x x^2\right )+\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log \left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )}{\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log ^2\left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )} \, dx=\frac {x}{\log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right )} \] Input:

integrate(((exp(exp(x))*exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x 
)))*log(exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x)+(-x*exp(exp(x))+exp(x)*x 
^2-x)*exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))-x*exp(exp(x)))/(exp(exp(x))*e 
xp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))/log(exp(((1+x)*exp(ex 
p(x))+x)/exp(exp(x)))+x)^2,x, algorithm="maxima")
 

Output:

x/log(x + e^(x*e^(-e^x) + x + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (18) = 36\).

Time = 0.22 (sec) , antiderivative size = 194, normalized size of antiderivative = 9.24 \[ \int \frac {-e^{e^x} x+e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )} \left (-x-e^{e^x} x+e^x x^2\right )+\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log \left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )}{\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log ^2\left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )} \, dx=\frac {x^{2} e^{\left (2 \, x e^{\left (-e^{x}\right )} + 3 \, x + 1\right )} - x e^{\left (2 \, x e^{\left (-e^{x}\right )} + 2 \, x + e^{x} + 1\right )} - x e^{\left (2 \, x e^{\left (-e^{x}\right )} + 2 \, x + 1\right )} - x e^{\left (x e^{\left (-e^{x}\right )} + x + e^{x}\right )}}{x e^{\left (2 \, x e^{\left (-e^{x}\right )} + 3 \, x + 1\right )} \log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right ) - e^{\left (2 \, x e^{\left (-e^{x}\right )} + 2 \, x + e^{x} + 1\right )} \log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right ) - e^{\left (2 \, x e^{\left (-e^{x}\right )} + 2 \, x + 1\right )} \log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right ) - e^{\left (x e^{\left (-e^{x}\right )} + x + e^{x}\right )} \log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right )} \] Input:

integrate(((exp(exp(x))*exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x 
)))*log(exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x)+(-x*exp(exp(x))+exp(x)*x 
^2-x)*exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))-x*exp(exp(x)))/(exp(exp(x))*e 
xp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))/log(exp(((1+x)*exp(ex 
p(x))+x)/exp(exp(x)))+x)^2,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.71 (sec) , antiderivative size = 451, normalized size of antiderivative = 21.48 \[ \int \frac {-e^{e^x} x+e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )} \left (-x-e^{e^x} x+e^x x^2\right )+\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log \left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )}{\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log ^2\left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )} \, dx =\text {Too large to display} \] Input:

int(-(x*exp(exp(x)) + exp(exp(-exp(x))*(x + exp(exp(x))*(x + 1)))*(x - x^2 
*exp(x) + x*exp(exp(x))) - log(x + exp(exp(-exp(x))*(x + exp(exp(x))*(x + 
1))))*(x*exp(exp(x)) + exp(exp(x))*exp(exp(-exp(x))*(x + exp(exp(x))*(x + 
1)))))/(log(x + exp(exp(-exp(x))*(x + exp(exp(x))*(x + 1))))^2*(x*exp(exp( 
x)) + exp(exp(x))*exp(exp(-exp(x))*(x + exp(exp(x))*(x + 1))))),x)
 

Output:

(x - (exp(exp(x))*log(x + exp(1)*exp(x)*exp(x*exp(-exp(x))))*(x + exp(x + 
x*exp(-exp(x)) + 1)))/(exp(exp(x)) - x*exp(2*x + x*exp(-exp(x)) + 1) + 2*e 
xp(x + exp(x)/2 + x*exp(-exp(x)) + 1)*cosh(exp(x)/2)))/log(x + exp(1)*exp( 
x)*exp(x*exp(-exp(x)))) - (exp(4*exp(x)) + 2*exp(5*exp(x)) + exp(6*exp(x)) 
 - exp(2*x + 4*exp(x))*(2*x^2 + 4*x^3) - 3*x^3*exp(2*x + 3*exp(x)) + x^3*e 
xp(3*x + 4*exp(x)) + x^4*exp(3*x + 3*exp(x)) + exp(2*x + 5*exp(x))*(x - x^ 
2) + 3*x^2*exp(x + 3*exp(x)) + 7*x^2*exp(x + 4*exp(x)) - exp(x + 5*exp(x)) 
*(x - 4*x^2 + 2) - 6*x*exp((9*exp(x))/2)*cosh(exp(x)/2) - 2*x*exp((9*exp(x 
))/2)*cosh((3*exp(x))/2))/((exp(exp(x)) + exp(x + x*exp(-exp(x)) + 1)*(2*e 
xp(exp(x)/2)*cosh(exp(x)/2) - x*exp(x)))*(2*exp(exp(x)/2)*cosh(exp(x)/2) - 
 x*exp(x))*(2*exp(3*exp(x)) - exp(x + 3*exp(x))*(3*x + 2) + x^2*exp(2*x + 
2*exp(x)) + 2*cosh(exp(x))*exp(3*exp(x)) - 2*x*exp(x + 2*exp(x)) + x*exp(2 
*x + 3*exp(x)))) - (exp(x)*(x*exp(x) - 1)*(2*x*sinh(x/2)*exp(x/2) - 2))/(( 
2*exp(exp(x)/2)*cosh(exp(x)/2) - x*exp(x))*(2*exp(x) - 2*x*sinh(x/2)*exp(( 
3*x)/2)))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {-e^{e^x} x+e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )} \left (-x-e^{e^x} x+e^x x^2\right )+\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log \left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )}{\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log ^2\left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )} \, dx=\frac {x}{\mathrm {log}\left (e^{\frac {e^{e^{x}} x +x}{e^{e^{x}}}} e +x \right )} \] Input:

int(((exp(exp(x))*exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))*lo 
g(exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))+x)+(-x*exp(exp(x))+exp(x)*x^2-x)* 
exp(((1+x)*exp(exp(x))+x)/exp(exp(x)))-x*exp(exp(x)))/(exp(exp(x))*exp(((1 
+x)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))/log(exp(((1+x)*exp(exp(x))+ 
x)/exp(exp(x)))+x)^2,x)
 

Output:

x/log(e**((e**(e**x)*x + x)/e**(e**x))*e + x)