\(\int \frac {-x^3 \log (x)+8 e^{e^x+x} x^3 \log (x)+8 e^{2 e^x+x} x^3 \log (x)+(x^2-x^3+(-2 x^2+2 x^3) \log (x)+e^{e^x} (8 x^2-16 x^2 \log (x))+e^{2 e^x} (4 x^2-8 x^2 \log (x))) \log (\frac {1}{4} (-1-8 e^{e^x}-4 e^{2 e^x}+x))+(2+16 e^{e^x}+8 e^{2 e^x}-2 x) \log ^2(\frac {1}{4} (-1-8 e^{e^x}-4 e^{2 e^x}+x))}{x^5+8 e^{e^x} x^5+4 e^{2 e^x} x^5-x^6+(4 x^3+32 e^{e^x} x^3+16 e^{2 e^x} x^3-4 x^4) \log (\frac {1}{4} (-1-8 e^{e^x}-4 e^{2 e^x}+x))+(4 x+32 e^{e^x} x+16 e^{2 e^x} x-4 x^2) \log ^2(\frac {1}{4} (-1-8 e^{e^x}-4 e^{2 e^x}+x))} \, dx\) [38]

Optimal result

Integrand size = 320, antiderivative size = 33 \[ \int \frac {-x^3 \log (x)+8 e^{e^x+x} x^3 \log (x)+8 e^{2 e^x+x} x^3 \log (x)+\left (x^2-x^3+\left (-2 x^2+2 x^3\right ) \log (x)+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (2+16 e^{e^x}+8 e^{2 e^x}-2 x\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )}{x^5+8 e^{e^x} x^5+4 e^{2 e^x} x^5-x^6+\left (4 x^3+32 e^{e^x} x^3+16 e^{2 e^x} x^3-4 x^4\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (4 x+32 e^{e^x} x+16 e^{2 e^x} x-4 x^2\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )} \, dx=\frac {\log (x)}{2+\frac {x^2}{\log \left (-\left (1+e^{e^x}\right )^2+\frac {3+x}{4}\right )}} \] Output:

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

Mathematica [F]

\[ \int \frac {-x^3 \log (x)+8 e^{e^x+x} x^3 \log (x)+8 e^{2 e^x+x} x^3 \log (x)+\left (x^2-x^3+\left (-2 x^2+2 x^3\right ) \log (x)+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (2+16 e^{e^x}+8 e^{2 e^x}-2 x\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )}{x^5+8 e^{e^x} x^5+4 e^{2 e^x} x^5-x^6+\left (4 x^3+32 e^{e^x} x^3+16 e^{2 e^x} x^3-4 x^4\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (4 x+32 e^{e^x} x+16 e^{2 e^x} x-4 x^2\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )} \, dx=\int \frac {-x^3 \log (x)+8 e^{e^x+x} x^3 \log (x)+8 e^{2 e^x+x} x^3 \log (x)+\left (x^2-x^3+\left (-2 x^2+2 x^3\right ) \log (x)+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (2+16 e^{e^x}+8 e^{2 e^x}-2 x\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )}{x^5+8 e^{e^x} x^5+4 e^{2 e^x} x^5-x^6+\left (4 x^3+32 e^{e^x} x^3+16 e^{2 e^x} x^3-4 x^4\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (4 x+32 e^{e^x} x+16 e^{2 e^x} x-4 x^2\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )} \, dx \] Input:

Integrate[(-(x^3*Log[x]) + 8*E^(E^x + x)*x^3*Log[x] + 8*E^(2*E^x + x)*x^3* 
Log[x] + (x^2 - x^3 + (-2*x^2 + 2*x^3)*Log[x] + E^E^x*(8*x^2 - 16*x^2*Log[ 
x]) + E^(2*E^x)*(4*x^2 - 8*x^2*Log[x]))*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) + 
x)/4] + (2 + 16*E^E^x + 8*E^(2*E^x) - 2*x)*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) 
 + x)/4]^2)/(x^5 + 8*E^E^x*x^5 + 4*E^(2*E^x)*x^5 - x^6 + (4*x^3 + 32*E^E^x 
*x^3 + 16*E^(2*E^x)*x^3 - 4*x^4)*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) + x)/4] + 
 (4*x + 32*E^E^x*x + 16*E^(2*E^x)*x - 4*x^2)*Log[(-1 - 8*E^E^x - 4*E^(2*E^ 
x) + x)/4]^2),x]
 

Output:

Integrate[(-(x^3*Log[x]) + 8*E^(E^x + x)*x^3*Log[x] + 8*E^(2*E^x + x)*x^3* 
Log[x] + (x^2 - x^3 + (-2*x^2 + 2*x^3)*Log[x] + E^E^x*(8*x^2 - 16*x^2*Log[ 
x]) + E^(2*E^x)*(4*x^2 - 8*x^2*Log[x]))*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) + 
x)/4] + (2 + 16*E^E^x + 8*E^(2*E^x) - 2*x)*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) 
 + x)/4]^2)/(x^5 + 8*E^E^x*x^5 + 4*E^(2*E^x)*x^5 - x^6 + (4*x^3 + 32*E^E^x 
*x^3 + 16*E^(2*E^x)*x^3 - 4*x^4)*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) + x)/4] + 
 (4*x + 32*E^E^x*x + 16*E^(2*E^x)*x - 4*x^2)*Log[(-1 - 8*E^E^x - 4*E^(2*E^ 
x) + x)/4]^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.

\(\displaystyle \int \frac {8 e^{x+e^x} x^3 \log (x)+8 e^{x+2 e^x} x^3 \log (x)-x^3 \log (x)+\left (-x^3+x^2+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )+\left (2 x^3-2 x^2\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )+\left (-2 x+16 e^{e^x}+8 e^{2 e^x}+2\right ) \log ^2\left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )}{-x^6+8 e^{e^x} x^5+4 e^{2 e^x} x^5+x^5+\left (-4 x^2+32 e^{e^x} x+16 e^{2 e^x} x+4 x\right ) \log ^2\left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )+\left (-4 x^4+32 e^{e^x} x^3+16 e^{2 e^x} x^3+4 x^3\right ) \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {8 e^{x+e^x} x^3 \log (x)+8 e^{x+2 e^x} x^3 \log (x)-x^3 \log (x)+\left (-x^3+x^2+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )+\left (2 x^3-2 x^2\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )+\left (-2 x+16 e^{e^x}+8 e^{2 e^x}+2\right ) \log ^2\left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )}{\left (-x+8 e^{e^x}+4 e^{2 e^x}+1\right ) x \left (x^2+2 \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {8 e^{x+e^x} \left (e^{e^x}+1\right ) x^2 \log (x)}{\left (-x+8 e^{e^x}+4 e^{2 e^x}+1\right ) \left (x^2+2 \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )\right )^2}-\frac {x^3 (-\log (x))+2 x^3 \log (x) \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )-x^3 \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )+x^2 \log (16) \log (x)+8 e^{e^x} x^2 \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )+4 e^{2 e^x} x^2 \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )-16 e^{e^x} x^2 \log (x) \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )-8 e^{2 e^x} x^2 \log (x) \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )+x^2 \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )-2 x^2 \log (x) \log \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )-2 x \log ^2\left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )+16 e^{e^x} \log ^2\left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )+8 e^{2 e^x} \log ^2\left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )+2 \log ^2\left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )}{x \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right ) \left (x^2+2 \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \left (\frac {\log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )}{x^3+2 x \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )}+\frac {x \log (x) \left (8 e^{x+e^x} x+8 e^{x+2 e^x} x-x-2 \left (-x+8 e^{e^x}+4 e^{2 e^x}\right ) \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )-2 \log \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )+\log (16)\right )}{\left (-x+8 e^{e^x}+4 e^{2 e^x}+1\right ) \left (x^2+2 \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )}{x \left (x^2+2 \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )\right )}+\frac {x \log (x) \left (8 e^{x+e^x} x+8 e^{x+2 e^x} x-x+2 x \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )-16 e^{e^x} \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )-8 e^{2 e^x} \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )-2 \log \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )+\log (16)\right )}{\left (-x+8 e^{e^x}+4 e^{2 e^x}+1\right ) \left (x^2+2 \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {\log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )}{x \left (x^2+2 \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )\right )}+\frac {x \log (x) \left (8 e^{x+e^x} x+8 e^{x+2 e^x} x-x+2 x \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )-16 e^{e^x} \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )-8 e^{2 e^x} \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )-2 \log \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )+\log (16)\right )}{\left (-x+8 e^{e^x}+4 e^{2 e^x}+1\right ) \left (x^2+2 \log \left (\frac {1}{4} \left (x-8 e^{e^x}-4 e^{2 e^x}-1\right )\right )\right )^2}\right )dx\)

Input:

Int[(-(x^3*Log[x]) + 8*E^(E^x + x)*x^3*Log[x] + 8*E^(2*E^x + x)*x^3*Log[x] 
 + (x^2 - x^3 + (-2*x^2 + 2*x^3)*Log[x] + E^E^x*(8*x^2 - 16*x^2*Log[x]) + 
E^(2*E^x)*(4*x^2 - 8*x^2*Log[x]))*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) + x)/4] 
+ (2 + 16*E^E^x + 8*E^(2*E^x) - 2*x)*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) + x)/ 
4]^2)/(x^5 + 8*E^E^x*x^5 + 4*E^(2*E^x)*x^5 - x^6 + (4*x^3 + 32*E^E^x*x^3 + 
 16*E^(2*E^x)*x^3 - 4*x^4)*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) + x)/4] + (4*x 
+ 32*E^E^x*x + 16*E^(2*E^x)*x - 4*x^2)*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) + x 
)/4]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18

\[\frac {\ln \left (x \right )}{2}-\frac {x^{2} \ln \left (x \right )}{2 \left (x^{2}+2 \ln \left (-{\mathrm e}^{2 \,{\mathrm e}^{x}}-2 \,{\mathrm e}^{{\mathrm e}^{x}}+\frac {x}{4}-\frac {1}{4}\right )\right )}\]

Input:

int(((8*exp(exp(x))^2+16*exp(exp(x))-2*x+2)*ln(-exp(exp(x))^2-2*exp(exp(x) 
)+1/4*x-1/4)^2+((-8*x^2*ln(x)+4*x^2)*exp(exp(x))^2+(-16*x^2*ln(x)+8*x^2)*e 
xp(exp(x))+(2*x^3-2*x^2)*ln(x)-x^3+x^2)*ln(-exp(exp(x))^2-2*exp(exp(x))+1/ 
4*x-1/4)+8*x^3*exp(x)*ln(x)*exp(exp(x))^2+8*x^3*exp(x)*ln(x)*exp(exp(x))-x 
^3*ln(x))/((16*x*exp(exp(x))^2+32*x*exp(exp(x))-4*x^2+4*x)*ln(-exp(exp(x)) 
^2-2*exp(exp(x))+1/4*x-1/4)^2+(16*x^3*exp(exp(x))^2+32*x^3*exp(exp(x))-4*x 
^4+4*x^3)*ln(-exp(exp(x))^2-2*exp(exp(x))+1/4*x-1/4)+4*x^5*exp(exp(x))^2+8 
*x^5*exp(exp(x))-x^6+x^5),x)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (28) = 56\).

Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.52 \[ \int \frac {-x^3 \log (x)+8 e^{e^x+x} x^3 \log (x)+8 e^{2 e^x+x} x^3 \log (x)+\left (x^2-x^3+\left (-2 x^2+2 x^3\right ) \log (x)+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (2+16 e^{e^x}+8 e^{2 e^x}-2 x\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )}{x^5+8 e^{e^x} x^5+4 e^{2 e^x} x^5-x^6+\left (4 x^3+32 e^{e^x} x^3+16 e^{2 e^x} x^3-4 x^4\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (4 x+32 e^{e^x} x+16 e^{2 e^x} x-4 x^2\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )} \, dx=\frac {\log \left (\frac {1}{4} \, {\left ({\left (x - 1\right )} e^{\left (2 \, x\right )} - 4 \, e^{\left (2 \, x + 2 \, e^{x}\right )} - 8 \, e^{\left (2 \, x + e^{x}\right )}\right )} e^{\left (-2 \, x\right )}\right ) \log \left (x\right )}{x^{2} + 2 \, \log \left (\frac {1}{4} \, {\left ({\left (x - 1\right )} e^{\left (2 \, x\right )} - 4 \, e^{\left (2 \, x + 2 \, e^{x}\right )} - 8 \, e^{\left (2 \, x + e^{x}\right )}\right )} e^{\left (-2 \, x\right )}\right )} \] Input:

integrate(((8*exp(exp(x))^2+16*exp(exp(x))-2*x+2)*log(-exp(exp(x))^2-2*exp 
(exp(x))+1/4*x-1/4)^2+((-8*x^2*log(x)+4*x^2)*exp(exp(x))^2+(-16*x^2*log(x) 
+8*x^2)*exp(exp(x))+(2*x^3-2*x^2)*log(x)-x^3+x^2)*log(-exp(exp(x))^2-2*exp 
(exp(x))+1/4*x-1/4)+8*x^3*exp(x)*log(x)*exp(exp(x))^2+8*x^3*exp(x)*log(x)* 
exp(exp(x))-x^3*log(x))/((16*x*exp(exp(x))^2+32*x*exp(exp(x))-4*x^2+4*x)*l 
og(-exp(exp(x))^2-2*exp(exp(x))+1/4*x-1/4)^2+(16*x^3*exp(exp(x))^2+32*x^3* 
exp(exp(x))-4*x^4+4*x^3)*log(-exp(exp(x))^2-2*exp(exp(x))+1/4*x-1/4)+4*x^5 
*exp(exp(x))^2+8*x^5*exp(exp(x))-x^6+x^5),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 1.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {-x^3 \log (x)+8 e^{e^x+x} x^3 \log (x)+8 e^{2 e^x+x} x^3 \log (x)+\left (x^2-x^3+\left (-2 x^2+2 x^3\right ) \log (x)+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (2+16 e^{e^x}+8 e^{2 e^x}-2 x\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )}{x^5+8 e^{e^x} x^5+4 e^{2 e^x} x^5-x^6+\left (4 x^3+32 e^{e^x} x^3+16 e^{2 e^x} x^3-4 x^4\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (4 x+32 e^{e^x} x+16 e^{2 e^x} x-4 x^2\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )} \, dx=- \frac {x^{2} \log {\left (x \right )}}{2 x^{2} + 4 \log {\left (\frac {x}{4} - e^{2 e^{x}} - 2 e^{e^{x}} - \frac {1}{4} \right )}} + \frac {\log {\left (x \right )}}{2} \] Input:

integrate(((8*exp(exp(x))**2+16*exp(exp(x))-2*x+2)*ln(-exp(exp(x))**2-2*ex 
p(exp(x))+1/4*x-1/4)**2+((-8*x**2*ln(x)+4*x**2)*exp(exp(x))**2+(-16*x**2*l 
n(x)+8*x**2)*exp(exp(x))+(2*x**3-2*x**2)*ln(x)-x**3+x**2)*ln(-exp(exp(x))* 
*2-2*exp(exp(x))+1/4*x-1/4)+8*x**3*exp(x)*ln(x)*exp(exp(x))**2+8*x**3*exp( 
x)*ln(x)*exp(exp(x))-x**3*ln(x))/((16*x*exp(exp(x))**2+32*x*exp(exp(x))-4* 
x**2+4*x)*ln(-exp(exp(x))**2-2*exp(exp(x))+1/4*x-1/4)**2+(16*x**3*exp(exp( 
x))**2+32*x**3*exp(exp(x))-4*x**4+4*x**3)*ln(-exp(exp(x))**2-2*exp(exp(x)) 
+1/4*x-1/4)+4*x**5*exp(exp(x))**2+8*x**5*exp(exp(x))-x**6+x**5),x)
 

Output:

-x**2*log(x)/(2*x**2 + 4*log(x/4 - exp(2*exp(x)) - 2*exp(exp(x)) - 1/4)) + 
 log(x)/2
 

Maxima [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {-x^3 \log (x)+8 e^{e^x+x} x^3 \log (x)+8 e^{2 e^x+x} x^3 \log (x)+\left (x^2-x^3+\left (-2 x^2+2 x^3\right ) \log (x)+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (2+16 e^{e^x}+8 e^{2 e^x}-2 x\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )}{x^5+8 e^{e^x} x^5+4 e^{2 e^x} x^5-x^6+\left (4 x^3+32 e^{e^x} x^3+16 e^{2 e^x} x^3-4 x^4\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (4 x+32 e^{e^x} x+16 e^{2 e^x} x-4 x^2\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )} \, dx=-\frac {x^{2} \log \left (x\right )}{2 \, {\left (x^{2} - 4 \, \log \left (2\right ) + 2 \, \log \left (x - 4 \, e^{\left (2 \, e^{x}\right )} - 8 \, e^{\left (e^{x}\right )} - 1\right )\right )}} + \frac {1}{2} \, \log \left (x\right ) \] Input:

integrate(((8*exp(exp(x))^2+16*exp(exp(x))-2*x+2)*log(-exp(exp(x))^2-2*exp 
(exp(x))+1/4*x-1/4)^2+((-8*x^2*log(x)+4*x^2)*exp(exp(x))^2+(-16*x^2*log(x) 
+8*x^2)*exp(exp(x))+(2*x^3-2*x^2)*log(x)-x^3+x^2)*log(-exp(exp(x))^2-2*exp 
(exp(x))+1/4*x-1/4)+8*x^3*exp(x)*log(x)*exp(exp(x))^2+8*x^3*exp(x)*log(x)* 
exp(exp(x))-x^3*log(x))/((16*x*exp(exp(x))^2+32*x*exp(exp(x))-4*x^2+4*x)*l 
og(-exp(exp(x))^2-2*exp(exp(x))+1/4*x-1/4)^2+(16*x^3*exp(exp(x))^2+32*x^3* 
exp(exp(x))-4*x^4+4*x^3)*log(-exp(exp(x))^2-2*exp(exp(x))+1/4*x-1/4)+4*x^5 
*exp(exp(x))^2+8*x^5*exp(exp(x))-x^6+x^5),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (28) = 56\).

Time = 0.63 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.73 \[ \int \frac {-x^3 \log (x)+8 e^{e^x+x} x^3 \log (x)+8 e^{2 e^x+x} x^3 \log (x)+\left (x^2-x^3+\left (-2 x^2+2 x^3\right ) \log (x)+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (2+16 e^{e^x}+8 e^{2 e^x}-2 x\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )}{x^5+8 e^{e^x} x^5+4 e^{2 e^x} x^5-x^6+\left (4 x^3+32 e^{e^x} x^3+16 e^{2 e^x} x^3-4 x^4\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (4 x+32 e^{e^x} x+16 e^{2 e^x} x-4 x^2\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )} \, dx=-\frac {2 \, \log \left (2\right ) \log \left (x\right ) - \log \left (x - 4 \, e^{\left (2 \, e^{x}\right )} - 8 \, e^{\left (e^{x}\right )} - 1\right ) \log \left (x\right )}{x^{2} - 4 \, \log \left (2\right ) + 2 \, \log \left (x - 4 \, e^{\left (2 \, e^{x}\right )} - 8 \, e^{\left (e^{x}\right )} - 1\right )} \] Input:

integrate(((8*exp(exp(x))^2+16*exp(exp(x))-2*x+2)*log(-exp(exp(x))^2-2*exp 
(exp(x))+1/4*x-1/4)^2+((-8*x^2*log(x)+4*x^2)*exp(exp(x))^2+(-16*x^2*log(x) 
+8*x^2)*exp(exp(x))+(2*x^3-2*x^2)*log(x)-x^3+x^2)*log(-exp(exp(x))^2-2*exp 
(exp(x))+1/4*x-1/4)+8*x^3*exp(x)*log(x)*exp(exp(x))^2+8*x^3*exp(x)*log(x)* 
exp(exp(x))-x^3*log(x))/((16*x*exp(exp(x))^2+32*x*exp(exp(x))-4*x^2+4*x)*l 
og(-exp(exp(x))^2-2*exp(exp(x))+1/4*x-1/4)^2+(16*x^3*exp(exp(x))^2+32*x^3* 
exp(exp(x))-4*x^4+4*x^3)*log(-exp(exp(x))^2-2*exp(exp(x))+1/4*x-1/4)+4*x^5 
*exp(exp(x))^2+8*x^5*exp(exp(x))-x^6+x^5),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-x^3 \log (x)+8 e^{e^x+x} x^3 \log (x)+8 e^{2 e^x+x} x^3 \log (x)+\left (x^2-x^3+\left (-2 x^2+2 x^3\right ) \log (x)+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (2+16 e^{e^x}+8 e^{2 e^x}-2 x\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )}{x^5+8 e^{e^x} x^5+4 e^{2 e^x} x^5-x^6+\left (4 x^3+32 e^{e^x} x^3+16 e^{2 e^x} x^3-4 x^4\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (4 x+32 e^{e^x} x+16 e^{2 e^x} x-4 x^2\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )} \, dx=\int \frac {{\ln \left (\frac {x}{4}-2\,{\mathrm {e}}^{{\mathrm {e}}^x}-{\mathrm {e}}^{2\,{\mathrm {e}}^x}-\frac {1}{4}\right )}^2\,\left (16\,{\mathrm {e}}^{{\mathrm {e}}^x}-2\,x+8\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\right )-\ln \left (\frac {x}{4}-2\,{\mathrm {e}}^{{\mathrm {e}}^x}-{\mathrm {e}}^{2\,{\mathrm {e}}^x}-\frac {1}{4}\right )\,\left (\ln \left (x\right )\,\left (2\,x^2-2\,x^3\right )+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (16\,x^2\,\ln \left (x\right )-8\,x^2\right )+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (8\,x^2\,\ln \left (x\right )-4\,x^2\right )-x^2+x^3\right )-x^3\,\ln \left (x\right )+8\,x^3\,{\mathrm {e}}^{x+{\mathrm {e}}^x}\,\ln \left (x\right )+8\,x^3\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^x}\,\ln \left (x\right )}{{\ln \left (\frac {x}{4}-2\,{\mathrm {e}}^{{\mathrm {e}}^x}-{\mathrm {e}}^{2\,{\mathrm {e}}^x}-\frac {1}{4}\right )}^2\,\left (4\,x+32\,x\,{\mathrm {e}}^{{\mathrm {e}}^x}+16\,x\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}-4\,x^2\right )+8\,x^5\,{\mathrm {e}}^{{\mathrm {e}}^x}+\ln \left (\frac {x}{4}-2\,{\mathrm {e}}^{{\mathrm {e}}^x}-{\mathrm {e}}^{2\,{\mathrm {e}}^x}-\frac {1}{4}\right )\,\left (32\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^x}+4\,x^3-4\,x^4+16\,x^3\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}\right )+x^5-x^6+4\,x^5\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}} \,d x \] Input:

int((log(x/4 - 2*exp(exp(x)) - exp(2*exp(x)) - 1/4)^2*(16*exp(exp(x)) - 2* 
x + 8*exp(2*exp(x)) + 2) - log(x/4 - 2*exp(exp(x)) - exp(2*exp(x)) - 1/4)* 
(log(x)*(2*x^2 - 2*x^3) + exp(exp(x))*(16*x^2*log(x) - 8*x^2) + exp(2*exp( 
x))*(8*x^2*log(x) - 4*x^2) - x^2 + x^3) - x^3*log(x) + 8*x^3*exp(exp(x))*e 
xp(x)*log(x) + 8*x^3*exp(2*exp(x))*exp(x)*log(x))/(log(x/4 - 2*exp(exp(x)) 
 - exp(2*exp(x)) - 1/4)^2*(4*x + 32*x*exp(exp(x)) + 16*x*exp(2*exp(x)) - 4 
*x^2) + 8*x^5*exp(exp(x)) + log(x/4 - 2*exp(exp(x)) - exp(2*exp(x)) - 1/4) 
*(32*x^3*exp(exp(x)) + 4*x^3 - 4*x^4 + 16*x^3*exp(2*exp(x))) + x^5 - x^6 + 
 4*x^5*exp(2*exp(x))),x)
 

Output:

int((log(x/4 - 2*exp(exp(x)) - exp(2*exp(x)) - 1/4)^2*(16*exp(exp(x)) - 2* 
x + 8*exp(2*exp(x)) + 2) - log(x/4 - 2*exp(exp(x)) - exp(2*exp(x)) - 1/4)* 
(log(x)*(2*x^2 - 2*x^3) + exp(exp(x))*(16*x^2*log(x) - 8*x^2) + exp(2*exp( 
x))*(8*x^2*log(x) - 4*x^2) - x^2 + x^3) - x^3*log(x) + 8*x^3*exp(x + exp(x 
))*log(x) + 8*x^3*exp(x + 2*exp(x))*log(x))/(log(x/4 - 2*exp(exp(x)) - exp 
(2*exp(x)) - 1/4)^2*(4*x + 32*x*exp(exp(x)) + 16*x*exp(2*exp(x)) - 4*x^2) 
+ 8*x^5*exp(exp(x)) + log(x/4 - 2*exp(exp(x)) - exp(2*exp(x)) - 1/4)*(32*x 
^3*exp(exp(x)) + 4*x^3 - 4*x^4 + 16*x^3*exp(2*exp(x))) + x^5 - x^6 + 4*x^5 
*exp(2*exp(x))), x)
 

Reduce [F]

\[ \int \frac {-x^3 \log (x)+8 e^{e^x+x} x^3 \log (x)+8 e^{2 e^x+x} x^3 \log (x)+\left (x^2-x^3+\left (-2 x^2+2 x^3\right ) \log (x)+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (2+16 e^{e^x}+8 e^{2 e^x}-2 x\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )}{x^5+8 e^{e^x} x^5+4 e^{2 e^x} x^5-x^6+\left (4 x^3+32 e^{e^x} x^3+16 e^{2 e^x} x^3-4 x^4\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (4 x+32 e^{e^x} x+16 e^{2 e^x} x-4 x^2\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )} \, dx=\int \frac {\left (8 \left ({\mathrm e}^{{\mathrm e}^{x}}\right )^{2}+16 \,{\mathrm e}^{{\mathrm e}^{x}}-2 x +2\right ) {\mathrm {log}\left (-\left ({\mathrm e}^{{\mathrm e}^{x}}\right )^{2}-2 \,{\mathrm e}^{{\mathrm e}^{x}}+\frac {x}{4}-\frac {1}{4}\right )}^{2}+\left (\left (-8 \,\mathrm {log}\left (x \right ) x^{2}+4 x^{2}\right ) \left ({\mathrm e}^{{\mathrm e}^{x}}\right )^{2}+\left (-16 \,\mathrm {log}\left (x \right ) x^{2}+8 x^{2}\right ) {\mathrm e}^{{\mathrm e}^{x}}+\left (2 x^{3}-2 x^{2}\right ) \mathrm {log}\left (x \right )-x^{3}+x^{2}\right ) \mathrm {log}\left (-\left ({\mathrm e}^{{\mathrm e}^{x}}\right )^{2}-2 \,{\mathrm e}^{{\mathrm e}^{x}}+\frac {x}{4}-\frac {1}{4}\right )+8 x^{3} {\mathrm e}^{x} \mathrm {log}\left (x \right ) \left ({\mathrm e}^{{\mathrm e}^{x}}\right )^{2}+8 x^{3} {\mathrm e}^{x} \mathrm {log}\left (x \right ) {\mathrm e}^{{\mathrm e}^{x}}-\mathrm {log}\left (x \right ) x^{3}}{\left (16 x \left ({\mathrm e}^{{\mathrm e}^{x}}\right )^{2}+32 x \,{\mathrm e}^{{\mathrm e}^{x}}-4 x^{2}+4 x \right ) {\mathrm {log}\left (-\left ({\mathrm e}^{{\mathrm e}^{x}}\right )^{2}-2 \,{\mathrm e}^{{\mathrm e}^{x}}+\frac {x}{4}-\frac {1}{4}\right )}^{2}+\left (16 x^{3} \left ({\mathrm e}^{{\mathrm e}^{x}}\right )^{2}+32 x^{3} {\mathrm e}^{{\mathrm e}^{x}}-4 x^{4}+4 x^{3}\right ) \mathrm {log}\left (-\left ({\mathrm e}^{{\mathrm e}^{x}}\right )^{2}-2 \,{\mathrm e}^{{\mathrm e}^{x}}+\frac {x}{4}-\frac {1}{4}\right )+4 x^{5} \left ({\mathrm e}^{{\mathrm e}^{x}}\right )^{2}+8 x^{5} {\mathrm e}^{{\mathrm e}^{x}}-x^{6}+x^{5}}d x \] Input:

int(((8*exp(exp(x))^2+16*exp(exp(x))-2*x+2)*log(-exp(exp(x))^2-2*exp(exp(x 
))+1/4*x-1/4)^2+((-8*x^2*log(x)+4*x^2)*exp(exp(x))^2+(-16*x^2*log(x)+8*x^2 
)*exp(exp(x))+(2*x^3-2*x^2)*log(x)-x^3+x^2)*log(-exp(exp(x))^2-2*exp(exp(x 
))+1/4*x-1/4)+8*x^3*exp(x)*log(x)*exp(exp(x))^2+8*x^3*exp(x)*log(x)*exp(ex 
p(x))-x^3*log(x))/((16*x*exp(exp(x))^2+32*x*exp(exp(x))-4*x^2+4*x)*log(-ex 
p(exp(x))^2-2*exp(exp(x))+1/4*x-1/4)^2+(16*x^3*exp(exp(x))^2+32*x^3*exp(ex 
p(x))-4*x^4+4*x^3)*log(-exp(exp(x))^2-2*exp(exp(x))+1/4*x-1/4)+4*x^5*exp(e 
xp(x))^2+8*x^5*exp(exp(x))-x^6+x^5),x)
 

Output:

int(((8*exp(exp(x))^2+16*exp(exp(x))-2*x+2)*log(-exp(exp(x))^2-2*exp(exp(x 
))+1/4*x-1/4)^2+((-8*x^2*log(x)+4*x^2)*exp(exp(x))^2+(-16*x^2*log(x)+8*x^2 
)*exp(exp(x))+(2*x^3-2*x^2)*log(x)-x^3+x^2)*log(-exp(exp(x))^2-2*exp(exp(x 
))+1/4*x-1/4)+8*x^3*exp(x)*log(x)*exp(exp(x))^2+8*x^3*exp(x)*log(x)*exp(ex 
p(x))-x^3*log(x))/((16*x*exp(exp(x))^2+32*x*exp(exp(x))-4*x^2+4*x)*log(-ex 
p(exp(x))^2-2*exp(exp(x))+1/4*x-1/4)^2+(16*x^3*exp(exp(x))^2+32*x^3*exp(ex 
p(x))-4*x^4+4*x^3)*log(-exp(exp(x))^2-2*exp(exp(x))+1/4*x-1/4)+4*x^5*exp(e 
xp(x))^2+8*x^5*exp(exp(x))-x^6+x^5),x)