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