\[ \int \frac {e^{-1+e^{x^2}} \left (-10+e^{x^2} \left (40 x-20 x^2+e^x \left (8 x-4 x^2\right )\right )\right )+e^{-1+e^{x^2}+x^2} \left (10 x+2 e^x x\right ) \log \left (25+10 e^x+e^{2 x}\right )}{5+e^x} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{{\mathrm e}^{x^{2}}-1} \left (\ln \! \left (\left ({\mathrm e}^{x}+5\right )^{2}\right )-2 x +4\right ) \]
command
Int[(E^(-1 + E^x^2)*(-10 + E^x^2*(40*x - 20*x^2 + E^x*(8*x - 4*x^2))) + E^(-1 + E^x^2 + x^2)*(10*x + 2*E^x*x)*Log[25 + 10*E^x + E^(2*x)])/(5 + E^x),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {e^{-1+e^{x^2}} \left (-10+e^{x^2} \left (40 x-20 x^2+e^x \left (8 x-4 x^2\right )\right )\right )+e^{-1+e^{x^2}+x^2} \left (10 x+2 e^x x\right ) \log \left (25+10 e^x+e^{2 x}\right )}{5+e^x} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {e^{-x^2+e^{x^2}-1} \left (10 e^{x^2} (2-x) x+2 e^{x^2+x} (2-x) x+e^{x^2} \left (e^x+5\right ) x \log \left (\left (e^x+5\right )^2\right )\right )}{\left (e^x+5\right ) x} \]