\[ \int \frac {e^{\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )} \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\frac {2 \left (x^{2}+x +\ln \left (\ln \left (x \right )+\frac {7 x}{8}\right )\right )^{2}}{9}} \]
command
Int[(E^((2*x^2 + 4*x^3 + 2*x^4 + (4*x + 4*x^2)*Log[(7*x + 8*Log[x])/8] + 2*Log[(7*x + 8*Log[x])/8]^2)/9)*(32*x + 60*x^2 + 56*x^3 + 84*x^4 + 56*x^5 + (32*x^2 + 96*x^3 + 64*x^4)*Log[x] + (32 + 28*x + 28*x^2 + 56*x^3 + (32*x + 64*x^2)*Log[x])*Log[(7*x + 8*Log[x])/8]))/(63*x^2 + 72*x*Log[x]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ e^{\frac {2}{9} \left (x^2+x+\log \left (\frac {7 x}{8}+\log (x)\right )\right )^2} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {\exp \left (\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )\right ) \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx \]________________________________________________________________________________________