\[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx \]
Optimal antiderivative \[ \frac {{\mathrm e} \,{\mathrm e}^{-x \ln \left (3\right )} x \ln \left (2\right )}{\left (\ln \left (x \right )-2\right ) \ln \left (x \right )} \]
command
Int[(2*E*Log[2] + (-4*E*Log[2] + 2*E*x*Log[2]*Log[3])*Log[x] + (E*Log[2] - E*x*Log[2]*Log[3])*Log[x]^2)/(4*3^x*Log[x]^2 - 4*3^x*Log[x]^3 + 3^x*Log[x]^4),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ -\frac {e 3^{-x} \log (2) \left (x \log (9) \log (x)-x \log (3) \log ^2(x)\right )}{\log (3) (2-\log (x))^2 \log ^2(x)} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx \]________________________________________________________________________________________