\[ \int \frac {e^{\frac {1}{16} \left (1+8 \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \log \left (\frac {e^5+e^x}{x}\right )} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\left (-\frac {1}{4}-\ln \left (2 \ln \left (\frac {{\mathrm e}^{5}+{\mathrm e}^{x}}{x}\right )\right )\right )^{2}} \]
command
integrate((((-4+4*x)*exp(x)-4*exp(5))*log(2*log((exp(5)+exp(x))/x))+(-1+x)*exp(x)-exp(5))*exp(log(2*log((exp(5)+exp(x))/x))^2+1/2*log(2*log((exp(5)+exp(x))/x))+1/16)/(2*exp(x)*x+2*x*exp(5))/log((exp(5)+exp(x))/x),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ e^{\left (\log \left (2 \, \log \left (\frac {e^{5}}{x} + \frac {e^{x}}{x}\right )\right )^{2} + \frac {1}{2} \, \log \left (2 \, \log \left (\frac {e^{5}}{x} + \frac {e^{x}}{x}\right )\right ) + \frac {1}{16}\right )} \]