\(\int \frac {e^{\frac {1}{\log (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} (e^4 x-x^2)}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}})}} (8-12 e^{2+e^{e^x}+x}+e^{3 e^{e^x}+3 x} (-e^6+2 e^2 x)+e^{2 e^{e^x}+2 x} (6 e^4-4 x-4 x^2-4 e^{e^x+x} x^2))}{(-8 x+12 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} (-6 e^4 x+2 x^2)+e^{3 e^{e^x}+3 x} (e^6 x-e^2 x^2)) \log ^2(\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} (e^4 x-x^2)}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}})} \, dx\) [629]

Optimal result

Integrand size = 308, antiderivative size = 35 \[ \int \frac {e^{\frac {1}{\log \left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )}} \left (8-12 e^{2+e^{e^x}+x}+e^{3 e^{e^x}+3 x} \left (-e^6+2 e^2 x\right )+e^{2 e^{e^x}+2 x} \left (6 e^4-4 x-4 x^2-4 e^{e^x+x} x^2\right )\right )}{\left (-8 x+12 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (-6 e^4 x+2 x^2\right )+e^{3 e^{e^x}+3 x} \left (e^6 x-e^2 x^2\right )\right ) \log ^2\left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )} \, dx=e^{\frac {1}{\log \left (x-\frac {x^2}{\left (-e^2+2 e^{-e^{e^x}-x}\right )^2}\right )}} \] Output:

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.74 \[ \int \frac {e^{\frac {1}{\log \left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )}} \left (8-12 e^{2+e^{e^x}+x}+e^{3 e^{e^x}+3 x} \left (-e^6+2 e^2 x\right )+e^{2 e^{e^x}+2 x} \left (6 e^4-4 x-4 x^2-4 e^{e^x+x} x^2\right )\right )}{\left (-8 x+12 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (-6 e^4 x+2 x^2\right )+e^{3 e^{e^x}+3 x} \left (e^6 x-e^2 x^2\right )\right ) \log ^2\left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )} \, dx=e^{\frac {1}{\log \left (\frac {x \left (4-4 e^{2+e^{e^x}+x}+e^{2 \left (2+e^{e^x}+x\right )}-e^{2 \left (e^{e^x}+x\right )} x\right )}{\left (-2+e^{2+e^{e^x}+x}\right )^2}\right )}} \] Input:

Integrate[(E^Log[(4*x - 4*E^(2 + E^E^x + x)*x + E^(2*E^E^x + 2*x)*(E^4*x - 
 x^2))/(4 - 4*E^(2 + E^E^x + x) + E^(4 + 2*E^E^x + 2*x))]^(-1)*(8 - 12*E^( 
2 + E^E^x + x) + E^(3*E^E^x + 3*x)*(-E^6 + 2*E^2*x) + E^(2*E^E^x + 2*x)*(6 
*E^4 - 4*x - 4*x^2 - 4*E^(E^x + x)*x^2)))/((-8*x + 12*E^(2 + E^E^x + x)*x 
+ E^(2*E^E^x + 2*x)*(-6*E^4*x + 2*x^2) + E^(3*E^E^x + 3*x)*(E^6*x - E^2*x^ 
2))*Log[(4*x - 4*E^(2 + E^E^x + x)*x + E^(2*E^E^x + 2*x)*(E^4*x - x^2))/(4 
 - 4*E^(2 + E^E^x + x) + E^(4 + 2*E^E^x + 2*x))]^2),x]
 

Output:

E^Log[(x*(4 - 4*E^(2 + E^E^x + x) + E^(2*(2 + E^E^x + x)) - E^(2*(E^E^x + 
x))*x))/(-2 + E^(2 + E^E^x + x))^2]^(-1)
 

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.

\(\displaystyle \int \frac {\left (e^{2 x+2 e^{e^x}} \left (-4 e^{x+e^x} x^2-4 x^2-4 x+6 e^4\right )+e^{3 x+3 e^{e^x}} \left (2 e^2 x-e^6\right )-12 e^{x+e^{e^x}+2}+8\right ) \exp \left (\frac {1}{\log \left (\frac {e^{2 x+2 e^{e^x}} \left (e^4 x-x^2\right )-4 e^{x+e^{e^x}+2} x+4 x}{-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4}\right )}\right )}{\left (e^{2 x+2 e^{e^x}} \left (2 x^2-6 e^4 x\right )+e^{3 x+3 e^{e^x}} \left (e^6 x-e^2 x^2\right )+12 e^{x+e^{e^x}+2} x-8 x\right ) \log ^2\left (\frac {e^{2 x+2 e^{e^x}} \left (e^4 x-x^2\right )-4 e^{x+e^{e^x}+2} x+4 x}{-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-e^{2 x+2 e^{e^x}} \left (-4 e^{x+e^x} x^2-4 x^2-4 x+6 e^4\right )-e^{3 x+3 e^{e^x}} \left (2 e^2 x-e^6\right )+12 e^{x+e^{e^x}+2}-8\right ) \exp \left (\frac {1}{\log \left (\frac {e^{2 x+2 e^{e^x}} \left (e^4 x-x^2\right )-4 e^{x+e^{e^x}+2} x+4 x}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {e^{2 x+2 e^{e^x}} \left (e^4 x-x^2\right )-4 e^{x+e^{e^x}+2} x+4 x}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (\frac {1}{\log \left (\frac {e^{2 x+2 e^{e^x}} \left (e^4 x-x^2\right )-4 e^{x+e^{e^x}+2} x+4 x}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (\frac {1}{\log \left (\frac {e^{2 x+2 e^{e^x}} \left (e^4 x-x^2\right )-4 e^{x+e^{e^x}+2} x+4 x}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {e^{2 x+2 e^{e^x}} \left (e^4 x-x^2\right )-4 e^{x+e^{e^x}+2} x+4 x}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (4 e^{3 x+2 e^{e^x}+e^x} x^2-2 e^{3 x+3 e^{e^x}+2} x+4 e^{2 \left (x+e^{e^x}\right )} (x+1) x+12 e^{x+e^{e^x}+2}-6 e^{2 \left (x+e^{e^x}+2\right )}+e^{3 \left (x+e^{e^x}+2\right )}-8\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )}{\left (2-e^{x+e^{e^x}+2}\right ) x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (4 e^{e^x} x^2-2 e^{e^{e^x}+2} x+e^{e^{e^x}+6}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^4-x\right ) x \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (2 e^{e^x}+e^{e^{e^x}+2}\right ) \exp \left (-e^{e^x}+\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-2\right )}{\left (e^{x+e^{e^x}+2}-2\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}+\frac {4 \left (-2 e^{e^{e^x}} x+e^{x+2 e^{e^x}+2} x-2 e^{x+e^{e^x}+e^x} x+4 e^{e^x+2}-2 e^{x+e^{e^x}+e^x+4}+\left (1-e^4\right ) e^{x+2 e^{e^x}+2}-\left (1-2 e^4\right ) e^{e^{e^x}}\right ) \exp \left (\frac {1}{\log \left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}-e^{e^x}\right )}{\left (e^4-x\right ) \left (-e^{2 x+2 e^{e^x}} x-4 e^{x+e^{e^x}+2}+e^{2 x+2 e^{e^x}+4}+4\right ) \log ^2\left (\frac {x \left (-e^{2 \left (x+e^{e^x}\right )} x-4 e^{x+e^{e^x}+2}+e^{2 \left (x+e^{e^x}+2\right )}+4\right )}{\left (e^{x+e^{e^x}+2}-2\right )^2}\right )}\right )dx\)

Input:

Int[(E^Log[(4*x - 4*E^(2 + E^E^x + x)*x + E^(2*E^E^x + 2*x)*(E^4*x - x^2)) 
/(4 - 4*E^(2 + E^E^x + x) + E^(4 + 2*E^E^x + 2*x))]^(-1)*(8 - 12*E^(2 + E^ 
E^x + x) + E^(3*E^E^x + 3*x)*(-E^6 + 2*E^2*x) + E^(2*E^E^x + 2*x)*(6*E^4 - 
 4*x - 4*x^2 - 4*E^(E^x + x)*x^2)))/((-8*x + 12*E^(2 + E^E^x + x)*x + E^(2 
*E^E^x + 2*x)*(-6*E^4*x + 2*x^2) + E^(3*E^E^x + 3*x)*(E^6*x - E^2*x^2))*Lo 
g[(4*x - 4*E^(2 + E^E^x + x)*x + E^(2*E^E^x + 2*x)*(E^4*x - x^2))/(4 - 4*E 
^(2 + E^E^x + x) + E^(4 + 2*E^E^x + 2*x))]^2),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.09 (sec) , antiderivative size = 828, normalized size of antiderivative = 23.66

\[\text {Expression too large to display}\]

Input:

int(((-exp(2)^3+2*exp(2)*x)*exp(x+exp(exp(x)))^3+(-4*x^2*exp(x)*exp(exp(x) 
)+6*exp(2)^2-4*x^2-4*x)*exp(x+exp(exp(x)))^2-12*exp(2)*exp(x+exp(exp(x)))+ 
8)*exp(1/ln(((x*exp(2)^2-x^2)*exp(x+exp(exp(x)))^2-4*x*exp(2)*exp(x+exp(ex 
p(x)))+4*x)/(exp(2)^2*exp(x+exp(exp(x)))^2-4*exp(2)*exp(x+exp(exp(x)))+4)) 
)/((x*exp(2)^3-x^2*exp(2))*exp(x+exp(exp(x)))^3+(-6*x*exp(2)^2+2*x^2)*exp( 
x+exp(exp(x)))^2+12*x*exp(2)*exp(x+exp(exp(x)))-8*x)/ln(((x*exp(2)^2-x^2)* 
exp(x+exp(exp(x)))^2-4*x*exp(2)*exp(x+exp(exp(x)))+4*x)/(exp(2)^2*exp(x+ex 
p(exp(x)))^2-4*exp(2)*exp(x+exp(exp(x)))+4))^2,x)
 

Output:

exp(2/(I*csgn(I*(exp(2*x+2*exp(exp(x)))*x+4*exp(2+x+exp(exp(x)))-exp(4+2*x 
+2*exp(exp(x)))-4)/(exp(2+x+exp(exp(x)))-2)^2)^3*Pi+I*csgn(I*(exp(2*x+2*ex 
p(exp(x)))*x+4*exp(2+x+exp(exp(x)))-exp(4+2*x+2*exp(exp(x)))-4)/(exp(2+x+e 
xp(exp(x)))-2)^2)^2*csgn(I/(exp(2+x+exp(exp(x)))-2)^2)*Pi-I*csgn(I*(exp(2* 
x+2*exp(exp(x)))*x+4*exp(2+x+exp(exp(x)))-exp(4+2*x+2*exp(exp(x)))-4)/(exp 
(2+x+exp(exp(x)))-2)^2)^2*Pi*csgn(I*(exp(2*x+2*exp(exp(x)))*x+4*exp(2+x+ex 
p(exp(x)))-exp(4+2*x+2*exp(exp(x)))-4))-I*csgn(I*(exp(2*x+2*exp(exp(x)))*x 
+4*exp(2+x+exp(exp(x)))-exp(4+2*x+2*exp(exp(x)))-4)/(exp(2+x+exp(exp(x)))- 
2)^2)*csgn(I/(exp(2+x+exp(exp(x)))-2)^2)*Pi*csgn(I*(exp(2*x+2*exp(exp(x))) 
*x+4*exp(2+x+exp(exp(x)))-exp(4+2*x+2*exp(exp(x)))-4))-I*csgn(I*(exp(2*x+2 
*exp(exp(x)))*x+4*exp(2+x+exp(exp(x)))-exp(4+2*x+2*exp(exp(x)))-4)/(exp(2+ 
x+exp(exp(x)))-2)^2)*Pi*csgn(I*x*(exp(2*x+2*exp(exp(x)))*x+4*exp(2+x+exp(e 
xp(x)))-exp(4+2*x+2*exp(exp(x)))-4)/(exp(2+x+exp(exp(x)))-2)^2)^2-I*csgn(I 
*(exp(2*x+2*exp(exp(x)))*x+4*exp(2+x+exp(exp(x)))-exp(4+2*x+2*exp(exp(x))) 
-4)/(exp(2+x+exp(exp(x)))-2)^2)*Pi*csgn(I*x*(exp(2*x+2*exp(exp(x)))*x+4*ex 
p(2+x+exp(exp(x)))-exp(4+2*x+2*exp(exp(x)))-4)/(exp(2+x+exp(exp(x)))-2)^2) 
*csgn(I*x)+I*Pi*csgn(I*x*(exp(2*x+2*exp(exp(x)))*x+4*exp(2+x+exp(exp(x)))- 
exp(4+2*x+2*exp(exp(x)))-4)/(exp(2+x+exp(exp(x)))-2)^2)^3+I*Pi*csgn(I*x*(e 
xp(2*x+2*exp(exp(x)))*x+4*exp(2+x+exp(exp(x)))-exp(4+2*x+2*exp(exp(x)))-4) 
/(exp(2+x+exp(exp(x)))-2)^2)^2*csgn(I*x)+I*Pi*csgn(I*(exp(2+x+exp(exp(x...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (28) = 56\).

Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.26 \[ \int \frac {e^{\frac {1}{\log \left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )}} \left (8-12 e^{2+e^{e^x}+x}+e^{3 e^{e^x}+3 x} \left (-e^6+2 e^2 x\right )+e^{2 e^{e^x}+2 x} \left (6 e^4-4 x-4 x^2-4 e^{e^x+x} x^2\right )\right )}{\left (-8 x+12 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (-6 e^4 x+2 x^2\right )+e^{3 e^{e^x}+3 x} \left (e^6 x-e^2 x^2\right )\right ) \log ^2\left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )} \, dx=e^{\left (\frac {1}{\log \left (\frac {4 \, x e^{4} - {\left (x^{2} - x e^{4}\right )} e^{\left (2 \, {\left ({\left (x + 2\right )} e^{x} + e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right )} - 4 \, x e^{\left ({\left ({\left (x + 2\right )} e^{x} + e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} + 4\right )}}{4 \, e^{4} + e^{\left (2 \, {\left ({\left (x + 2\right )} e^{x} + e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} + 4\right )} - 4 \, e^{\left ({\left ({\left (x + 2\right )} e^{x} + e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} + 4\right )}}\right )}\right )} \] Input:

integrate(((-exp(2)^3+2*exp(2)*x)*exp(x+exp(exp(x)))^3+(-4*x^2*exp(x)*exp( 
exp(x))+6*exp(2)^2-4*x^2-4*x)*exp(x+exp(exp(x)))^2-12*exp(2)*exp(x+exp(exp 
(x)))+8)*exp(1/log(((x*exp(2)^2-x^2)*exp(x+exp(exp(x)))^2-4*x*exp(2)*exp(x 
+exp(exp(x)))+4*x)/(exp(2)^2*exp(x+exp(exp(x)))^2-4*exp(2)*exp(x+exp(exp(x 
)))+4)))/((x*exp(2)^3-x^2*exp(2))*exp(x+exp(exp(x)))^3+(-6*x*exp(2)^2+2*x^ 
2)*exp(x+exp(exp(x)))^2+12*x*exp(2)*exp(x+exp(exp(x)))-8*x)/log(((x*exp(2) 
^2-x^2)*exp(x+exp(exp(x)))^2-4*x*exp(2)*exp(x+exp(exp(x)))+4*x)/(exp(2)^2* 
exp(x+exp(exp(x)))^2-4*exp(2)*exp(x+exp(exp(x)))+4))^2,x, algorithm="frica 
s")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{\log \left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )}} \left (8-12 e^{2+e^{e^x}+x}+e^{3 e^{e^x}+3 x} \left (-e^6+2 e^2 x\right )+e^{2 e^{e^x}+2 x} \left (6 e^4-4 x-4 x^2-4 e^{e^x+x} x^2\right )\right )}{\left (-8 x+12 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (-6 e^4 x+2 x^2\right )+e^{3 e^{e^x}+3 x} \left (e^6 x-e^2 x^2\right )\right ) \log ^2\left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )} \, dx=\text {Timed out} \] Input:

integrate(((-exp(2)**3+2*exp(2)*x)*exp(x+exp(exp(x)))**3+(-4*x**2*exp(x)*e 
xp(exp(x))+6*exp(2)**2-4*x**2-4*x)*exp(x+exp(exp(x)))**2-12*exp(2)*exp(x+e 
xp(exp(x)))+8)*exp(1/ln(((x*exp(2)**2-x**2)*exp(x+exp(exp(x)))**2-4*x*exp( 
2)*exp(x+exp(exp(x)))+4*x)/(exp(2)**2*exp(x+exp(exp(x)))**2-4*exp(2)*exp(x 
+exp(exp(x)))+4)))/((x*exp(2)**3-x**2*exp(2))*exp(x+exp(exp(x)))**3+(-6*x* 
exp(2)**2+2*x**2)*exp(x+exp(exp(x)))**2+12*x*exp(2)*exp(x+exp(exp(x)))-8*x 
)/ln(((x*exp(2)**2-x**2)*exp(x+exp(exp(x)))**2-4*x*exp(2)*exp(x+exp(exp(x) 
))+4*x)/(exp(2)**2*exp(x+exp(exp(x)))**2-4*exp(2)*exp(x+exp(exp(x)))+4))** 
2,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1058 vs. \(2 (28) = 56\).

Time = 4.26 (sec) , antiderivative size = 1058, normalized size of antiderivative = 30.23 \[ \int \frac {e^{\frac {1}{\log \left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )}} \left (8-12 e^{2+e^{e^x}+x}+e^{3 e^{e^x}+3 x} \left (-e^6+2 e^2 x\right )+e^{2 e^{e^x}+2 x} \left (6 e^4-4 x-4 x^2-4 e^{e^x+x} x^2\right )\right )}{\left (-8 x+12 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (-6 e^4 x+2 x^2\right )+e^{3 e^{e^x}+3 x} \left (e^6 x-e^2 x^2\right )\right ) \log ^2\left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )} \, dx=\text {Too large to display} \] Input:

integrate(((-exp(2)^3+2*exp(2)*x)*exp(x+exp(exp(x)))^3+(-4*x^2*exp(x)*exp( 
exp(x))+6*exp(2)^2-4*x^2-4*x)*exp(x+exp(exp(x)))^2-12*exp(2)*exp(x+exp(exp 
(x)))+8)*exp(1/log(((x*exp(2)^2-x^2)*exp(x+exp(exp(x)))^2-4*x*exp(2)*exp(x 
+exp(exp(x)))+4*x)/(exp(2)^2*exp(x+exp(exp(x)))^2-4*exp(2)*exp(x+exp(exp(x 
)))+4)))/((x*exp(2)^3-x^2*exp(2))*exp(x+exp(exp(x)))^3+(-6*x*exp(2)^2+2*x^ 
2)*exp(x+exp(exp(x)))^2+12*x*exp(2)*exp(x+exp(exp(x)))-8*x)/log(((x*exp(2) 
^2-x^2)*exp(x+exp(exp(x)))^2-4*x*exp(2)*exp(x+exp(exp(x)))+4*x)/(exp(2)^2* 
exp(x+exp(exp(x)))^2-4*exp(2)*exp(x+exp(exp(x)))+4))^2,x, algorithm="maxim 
a")
 

Output:

-4*x^2*e^(3*x + 1/(log(-(x - e^4)*e^(2*x + 2*e^(e^x)) - 4*e^(x + e^(e^x) + 
 2) + 4) + log(x) - 2*log(e^(x + e^(e^x) + 2) - 2)) + e^x + 2*e^(e^x))/((2 
*x*e^2 - e^6)*e^(3*x + 3*e^(e^x)) - 2*(2*x^2*e^(3*x + e^x) + (2*x^2 + 2*x 
- 3*e^4)*e^(2*x))*e^(2*e^(e^x)) - 12*e^(x + e^(e^x) + 2) + 8) - 4*x^2*e^(2 
*x + 1/(log(-(x - e^4)*e^(2*x + 2*e^(e^x)) - 4*e^(x + e^(e^x) + 2) + 4) + 
log(x) - 2*log(e^(x + e^(e^x) + 2) - 2)) + 2*e^(e^x))/((2*x*e^2 - e^6)*e^( 
3*x + 3*e^(e^x)) - 2*(2*x^2*e^(3*x + e^x) + (2*x^2 + 2*x - 3*e^4)*e^(2*x)) 
*e^(2*e^(e^x)) - 12*e^(x + e^(e^x) + 2) + 8) + 2*x*e^(3*x + 1/(log(-(x - e 
^4)*e^(2*x + 2*e^(e^x)) - 4*e^(x + e^(e^x) + 2) + 4) + log(x) - 2*log(e^(x 
 + e^(e^x) + 2) - 2)) + 3*e^(e^x) + 2)/((2*x*e^2 - e^6)*e^(3*x + 3*e^(e^x) 
) - 2*(2*x^2*e^(3*x + e^x) + (2*x^2 + 2*x - 3*e^4)*e^(2*x))*e^(2*e^(e^x)) 
- 12*e^(x + e^(e^x) + 2) + 8) - 4*x*e^(2*x + 1/(log(-(x - e^4)*e^(2*x + 2* 
e^(e^x)) - 4*e^(x + e^(e^x) + 2) + 4) + log(x) - 2*log(e^(x + e^(e^x) + 2) 
 - 2)) + 2*e^(e^x))/((2*x*e^2 - e^6)*e^(3*x + 3*e^(e^x)) - 2*(2*x^2*e^(3*x 
 + e^x) + (2*x^2 + 2*x - 3*e^4)*e^(2*x))*e^(2*e^(e^x)) - 12*e^(x + e^(e^x) 
 + 2) + 8) - e^(3*x + 1/(log(-(x - e^4)*e^(2*x + 2*e^(e^x)) - 4*e^(x + e^( 
e^x) + 2) + 4) + log(x) - 2*log(e^(x + e^(e^x) + 2) - 2)) + 3*e^(e^x) + 6) 
/((2*x*e^2 - e^6)*e^(3*x + 3*e^(e^x)) - 2*(2*x^2*e^(3*x + e^x) + (2*x^2 + 
2*x - 3*e^4)*e^(2*x))*e^(2*e^(e^x)) - 12*e^(x + e^(e^x) + 2) + 8) + 6*e^(2 
*x + 1/(log(-(x - e^4)*e^(2*x + 2*e^(e^x)) - 4*e^(x + e^(e^x) + 2) + 4)...
 

Giac [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{\log \left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )}} \left (8-12 e^{2+e^{e^x}+x}+e^{3 e^{e^x}+3 x} \left (-e^6+2 e^2 x\right )+e^{2 e^{e^x}+2 x} \left (6 e^4-4 x-4 x^2-4 e^{e^x+x} x^2\right )\right )}{\left (-8 x+12 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (-6 e^4 x+2 x^2\right )+e^{3 e^{e^x}+3 x} \left (e^6 x-e^2 x^2\right )\right ) \log ^2\left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )} \, dx=\text {Timed out} \] Input:

integrate(((-exp(2)^3+2*exp(2)*x)*exp(x+exp(exp(x)))^3+(-4*x^2*exp(x)*exp( 
exp(x))+6*exp(2)^2-4*x^2-4*x)*exp(x+exp(exp(x)))^2-12*exp(2)*exp(x+exp(exp 
(x)))+8)*exp(1/log(((x*exp(2)^2-x^2)*exp(x+exp(exp(x)))^2-4*x*exp(2)*exp(x 
+exp(exp(x)))+4*x)/(exp(2)^2*exp(x+exp(exp(x)))^2-4*exp(2)*exp(x+exp(exp(x 
)))+4)))/((x*exp(2)^3-x^2*exp(2))*exp(x+exp(exp(x)))^3+(-6*x*exp(2)^2+2*x^ 
2)*exp(x+exp(exp(x)))^2+12*x*exp(2)*exp(x+exp(exp(x)))-8*x)/log(((x*exp(2) 
^2-x^2)*exp(x+exp(exp(x)))^2-4*x*exp(2)*exp(x+exp(exp(x)))+4*x)/(exp(2)^2* 
exp(x+exp(exp(x)))^2-4*exp(2)*exp(x+exp(exp(x)))+4))^2,x, algorithm="giac" 
)
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 1.66 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.17 \[ \int \frac {e^{\frac {1}{\log \left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )}} \left (8-12 e^{2+e^{e^x}+x}+e^{3 e^{e^x}+3 x} \left (-e^6+2 e^2 x\right )+e^{2 e^{e^x}+2 x} \left (6 e^4-4 x-4 x^2-4 e^{e^x+x} x^2\right )\right )}{\left (-8 x+12 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (-6 e^4 x+2 x^2\right )+e^{3 e^{e^x}+3 x} \left (e^6 x-e^2 x^2\right )\right ) \log ^2\left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )} \, dx={\mathrm {e}}^{\frac {1}{\ln \left (\frac {4\,x-x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^4-4\,x\,{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^x}{{\mathrm {e}}^{2\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^4-4\,{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^x+4}\right )}} \] Input:

int((exp(1/log((4*x + exp(2*x + 2*exp(exp(x)))*(x*exp(4) - x^2) - 4*x*exp( 
2)*exp(x + exp(exp(x))))/(exp(2*x + 2*exp(exp(x)))*exp(4) - 4*exp(2)*exp(x 
 + exp(exp(x))) + 4)))*(exp(2*x + 2*exp(exp(x)))*(4*x - 6*exp(4) + 4*x^2 + 
 4*x^2*exp(exp(x))*exp(x)) + 12*exp(2)*exp(x + exp(exp(x))) + exp(3*x + 3* 
exp(exp(x)))*(exp(6) - 2*x*exp(2)) - 8))/(log((4*x + exp(2*x + 2*exp(exp(x 
)))*(x*exp(4) - x^2) - 4*x*exp(2)*exp(x + exp(exp(x))))/(exp(2*x + 2*exp(e 
xp(x)))*exp(4) - 4*exp(2)*exp(x + exp(exp(x))) + 4))^2*(8*x + exp(2*x + 2* 
exp(exp(x)))*(6*x*exp(4) - 2*x^2) - exp(3*x + 3*exp(exp(x)))*(x*exp(6) - x 
^2*exp(2)) - 12*x*exp(2)*exp(x + exp(exp(x))))),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 1.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.74 \[ \int \frac {e^{\frac {1}{\log \left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )}} \left (8-12 e^{2+e^{e^x}+x}+e^{3 e^{e^x}+3 x} \left (-e^6+2 e^2 x\right )+e^{2 e^{e^x}+2 x} \left (6 e^4-4 x-4 x^2-4 e^{e^x+x} x^2\right )\right )}{\left (-8 x+12 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (-6 e^4 x+2 x^2\right )+e^{3 e^{e^x}+3 x} \left (e^6 x-e^2 x^2\right )\right ) \log ^2\left (\frac {4 x-4 e^{2+e^{e^x}+x} x+e^{2 e^{e^x}+2 x} \left (e^4 x-x^2\right )}{4-4 e^{2+e^{e^x}+x}+e^{4+2 e^{e^x}+2 x}}\right )} \, dx=e^{\frac {1}{\mathrm {log}\left (\frac {e^{2 e^{e^{x}}+2 x} e^{4} x -e^{2 e^{e^{x}}+2 x} x^{2}-4 e^{e^{e^{x}}+x} e^{2} x +4 x}{e^{2 e^{e^{x}}+2 x} e^{4}-4 e^{e^{e^{x}}+x} e^{2}+4}\right )}} \] Input:

int(((-exp(2)^3+2*exp(2)*x)*exp(x+exp(exp(x)))^3+(-4*x^2*exp(x)*exp(exp(x) 
)+6*exp(2)^2-4*x^2-4*x)*exp(x+exp(exp(x)))^2-12*exp(2)*exp(x+exp(exp(x)))+ 
8)*exp(1/log(((x*exp(2)^2-x^2)*exp(x+exp(exp(x)))^2-4*x*exp(2)*exp(x+exp(e 
xp(x)))+4*x)/(exp(2)^2*exp(x+exp(exp(x)))^2-4*exp(2)*exp(x+exp(exp(x)))+4) 
))/((x*exp(2)^3-x^2*exp(2))*exp(x+exp(exp(x)))^3+(-6*x*exp(2)^2+2*x^2)*exp 
(x+exp(exp(x)))^2+12*x*exp(2)*exp(x+exp(exp(x)))-8*x)/log(((x*exp(2)^2-x^2 
)*exp(x+exp(exp(x)))^2-4*x*exp(2)*exp(x+exp(exp(x)))+4*x)/(exp(2)^2*exp(x+ 
exp(exp(x)))^2-4*exp(2)*exp(x+exp(exp(x)))+4))^2,x)
 

Output:

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