\[ \int \frac {\left (-x+2 x \log (x)+\left (1+e^x+2 x\right ) \log ^2(x)\right ) \sqrt {\frac {x^4+\left (2 x^2+2 e^x x^2+2 x^3+2 x^4\right ) \log (x)+\left (1+e^{2 x}+2 x+3 x^2+2 x^3+x^4+e^x \left (2+2 x+2 x^2\right )\right ) \log ^2(x)}{\log ^2(x)}}}{x^2 \log (x)+\left (1+e^x+x+x^2\right ) \log ^2(x)} \, dx \]
Optimal antiderivative \[ \sqrt {\left (-1-x -\frac {x^{2}}{\ln \! \left (x \right )}-x^{2}-{\mathrm e}^{x}\right )^{2}} \]
command
integrate(((exp(x)+2*x+1)*log(x)^2+2*x*log(x)-x)*(((exp(x)^2+(2*x^2+2*x+2)*exp(x)+x^4+2*x^3+3*x^2+2*x+1)*log(x)^2+(2*exp(x)*x^2+2*x^4+2*x^3+2*x^2)*log(x)+x^4)/log(x)^2)^(1/2)/((exp(x)+x^2+x+1)*log(x)^2+x^2*log(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
\[ -\frac {1}{3} \, \log \left (x\right )^{3} + \frac {3}{2} \, x^{2} - \frac {1}{2} \, \log \left (x\right )^{2} + 2 \, x + \frac {x^{2}}{\log \left (x\right )} + e^{x} \]