\[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx \]
Optimal antiderivative \[ 1-\frac {1}{\left ({\mathrm e}-3-x \right ) x \left (2 x -3-\frac {3}{x}\right )}+\ln \! \left (\ln \! \left (x \right )\right ) \]
command
Int[(81 + 216*x + 90*x^2 - 108*x^3 - 39*x^4 + 12*x^5 + 4*x^6 + E^2*(9 + 18*x - 3*x^2 - 12*x^3 + 4*x^4) + E*(-54 - 126*x - 18*x^2 + 78*x^3 - 8*x^5) + (12*x - 6*x^2 - 6*x^3 + E*(-3*x + 4*x^2))*Log[x])/((81*x + 216*x^2 + 90*x^3 - 108*x^4 - 39*x^5 + 12*x^6 + 4*x^7 + E^2*(9*x + 18*x^2 - 3*x^3 - 12*x^4 + 4*x^5) + E*(-54*x - 126*x^2 - 18*x^3 + 78*x^4 - 8*x^6))*Log[x]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \log (\log (x))-\frac {1}{(x-e+3) \left (-2 x^2+3 x+3\right )} \]