\[ \int \frac {5 x-x^3+e^2 \left (5-x^2\right )+e^{2 e^{2 e^{\log (x) \log \left (e^2+x\right )}}} \left (e^2+x+e^{2 e^{\log (x) \log \left (e^2+x\right )}+\log (x) \log \left (e^2+x\right )} \left (-4 x \log (x)+\left (-4 e^2-4 x\right ) \log \left (e^2+x\right )\right )\right )}{25 x-10 x^2+11 x^3-2 x^4+x^5+e^{4 e^{2 e^{\log (x) \log \left (e^2+x\right )}}} \left (e^2+x\right )+e^2 \left (25-10 x+11 x^2-2 x^3+x^4\right )+e^{2 e^{2 e^{\log (x) \log \left (e^2+x\right )}}} \left (10 x-2 x^2+2 x^3+e^2 \left (10-2 x+2 x^2\right )\right )} \, dx \]
Optimal antiderivative \[ \frac {x}{x^{2}-x +{\mathrm e}^{2 \,{\mathrm e}^{2 \,{\mathrm e}^{\ln \left (x \right ) \ln \left (x +{\mathrm e}^{2}\right )}}}+5} \]
command
Integrate[(5*x - x^3 + E^2*(5 - x^2) + E^(2*E^(2*E^(Log[x]*Log[E^2 + x])))*(E^2 + x + E^(2*E^(Log[x]*Log[E^2 + x]) + Log[x]*Log[E^2 + x])*(-4*x*Log[x] + (-4*E^2 - 4*x)*Log[E^2 + x])))/(25*x - 10*x^2 + 11*x^3 - 2*x^4 + x^5 + E^(4*E^(2*E^(Log[x]*Log[E^2 + x])))*(E^2 + x) + E^2*(25 - 10*x + 11*x^2 - 2*x^3 + x^4) + E^(2*E^(2*E^(Log[x]*Log[E^2 + x])))*(10*x - 2*x^2 + 2*x^3 + E^2*(10 - 2*x + 2*x^2))),x]
Mathematica 13.1 output
\[ \int \frac {5 x-x^3+e^2 \left (5-x^2\right )+e^{2 e^{2 e^{\log (x) \log \left (e^2+x\right )}}} \left (e^2+x+e^{2 e^{\log (x) \log \left (e^2+x\right )}+\log (x) \log \left (e^2+x\right )} \left (-4 x \log (x)+\left (-4 e^2-4 x\right ) \log \left (e^2+x\right )\right )\right )}{25 x-10 x^2+11 x^3-2 x^4+x^5+e^{4 e^{2 e^{\log (x) \log \left (e^2+x\right )}}} \left (e^2+x\right )+e^2 \left (25-10 x+11 x^2-2 x^3+x^4\right )+e^{2 e^{2 e^{\log (x) \log \left (e^2+x\right )}}} \left (10 x-2 x^2+2 x^3+e^2 \left (10-2 x+2 x^2\right )\right )} \, dx \]
Mathematica 12.3 output
\[ \frac {x \left (4 e^{2 x^{\log \left (e^2+x\right )}+\log (x) \log \left (e^2+x\right )} x \left (5-x+x^2\right ) \log (x)+\left (e^2+x\right ) \left (x-2 x^2+4 e^{2 x^{\log \left (e^2+x\right )}+\log (x) \log \left (e^2+x\right )} \left (5-x+x^2\right ) \log \left (e^2+x\right )\right )\right )}{\left (5+e^{2 e^{2 x^{\log \left (e^2+x\right )}}}-x+x^2\right ) \left (4 e^{2 x^{\log \left (e^2+x\right )}} x^{1+\log \left (e^2+x\right )} \left (5-x+x^2\right ) \log (x)+\left (e^2+x\right ) \left (x-2 x^2+4 e^{2 x^{\log \left (e^2+x\right )}} x^{\log \left (e^2+x\right )} \left (5-x+x^2\right ) \log \left (e^2+x\right )\right )\right )} \]