\[ \int \frac {e^{\frac {-x+\left (5+x^3\right ) \log \left (e^{e^{6 x}} x\right )}{\log \left (e^{e^{6 x}} x\right )}} \left (1+6 e^{6 x} x-\log \left (e^{e^{6 x}} x\right )+3 x^2 \log ^2\left (e^{e^{6 x}} x\right )\right )}{\log ^2\left (e^{e^{6 x}} x\right )} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{5-\frac {x}{\ln \left (x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right )}+x^{3}} \]
command
Int[(E^((-x + (5 + x^3)*Log[E^E^(6*x)*x])/Log[E^E^(6*x)*x])*(1 + 6*E^(6*x)*x - Log[E^E^(6*x)*x] + 3*x^2*Log[E^E^(6*x)*x]^2))/Log[E^E^(6*x)*x]^2,x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\exp \left (\frac {-x+\left (5+x^3\right ) \log \left (e^{e^{6 x}} x\right )}{\log \left (e^{e^{6 x}} x\right )}\right ) \left (1+6 e^{6 x} x-\log \left (e^{e^{6 x}} x\right )+3 x^2 \log ^2\left (e^{e^{6 x}} x\right )\right )}{\log ^2\left (e^{e^{6 x}} x\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ e^{x^3-\frac {x}{\log \left (e^{e^{6 x}} x\right )}+5} \]