\[ \int \frac {(-3-3 \log (x)) \log \left (\frac {3+16 x \log \left (-2+e^5\right ) \log (x)}{4 x \log \left (-2+e^5\right ) \log (x)}\right )}{6 x \log (x)+32 x^2 \log \left (-2+e^5\right ) \log ^2(x)} \, dx \]
Optimal antiderivative \[ \frac {{\ln \! \left (4+\frac {3}{4 x \ln \left (x \right ) \ln \left ({\mathrm e}^{5}-2\right )}\right )}^{2}}{4} \]
command
Int[((-3 - 3*Log[x])*Log[(3 + 16*x*Log[-2 + E^5]*Log[x])/(4*x*Log[-2 + E^5]*Log[x])])/(6*x*Log[x] + 32*x^2*Log[-2 + E^5]*Log[x]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {(-3-3 \log (x)) \log \left (\frac {3+16 x \log \left (-2+e^5\right ) \log (x)}{4 x \log \left (-2+e^5\right ) \log (x)}\right )}{6 x \log (x)+32 x^2 \log \left (-2+e^5\right ) \log ^2(x)} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {1}{4} \log ^2\left (\frac {3}{4 x \log \left (e^5-2\right ) \log (x)}+4\right ) \]