\[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {\ln \! \left (x^{2}-{\mathrm e}^{{\mathrm e}^{\frac {5 x}{{\mathrm e}^{3}+5}+5 x^{4}}}\right )}{x} \]
command
Int[(-10*x^2 - 2*E^3*x^2 + E^(E^((5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3)) + (5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3))*(5*x + 100*x^4 + 20*E^3*x^4) + (E^E^((5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3))*(-5 - E^3) + 5*x^2 + E^3*x^2)*Log[-E^E^((5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3)) + x^2])/(-5*x^4 - E^3*x^4 + E^E^((5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3))*(5*x^2 + E^3*x^2)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {-10 x^2-2 e^3 x^2+\exp \left (e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}\right ) \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {\log \left (x^2-e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}}\right )}{x} \]