\[ \int \frac {-\frac {3}{5} e^{2 x} \log (x)+\left (\frac {e^{2 x}}{5}+\frac {2}{5} e^{2 x} x \log (x)\right ) \log \left (x^3\right )}{5 x \log ^2\left (x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (x \right ) {\mathrm e}^{-\ln \left (5\right )+2 x}}{5 \ln \left (x^{3}\right )} \]
command
Int[((-3*E^(2*x)*Log[x])/5 + (E^(2*x)/5 + (2*E^(2*x)*x*Log[x])/5)*Log[x^3])/(5*x*Log[x^3]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \frac {e^{2 x} \log (x)}{25 \log \left (x^3\right )} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {-\frac {3}{5} e^{2 x} \log (x)+\left (\frac {e^{2 x}}{5}+\frac {2}{5} e^{2 x} x \log (x)\right ) \log \left (x^3\right )}{5 x \log ^2\left (x^3\right )} \, dx \]________________________________________________________________________________________