\[ \int \frac {20 x^2+24 x^3+9 x^4+x^5+\left (12+20 x+221 x^2+208 x^3+71 x^4+8 x^5\right ) \log (x)+\left (60 x^2+76 x^3+31 x^4+4 x^5\right ) \log (x) \log (\log (x))}{\left (4+4 x+x^2\right ) \log (x)} \, dx \]
Optimal antiderivative \[ x \left (3+x^{2} \left (5+x \right ) \left (2+\ln \! \left (\ln \! \left (x \right )\right )+\frac {3}{2+x}\right )+x \right ) \]
command
Int[(20*x^2 + 24*x^3 + 9*x^4 + x^5 + (12 + 20*x + 221*x^2 + 208*x^3 + 71*x^4 + 8*x^5)*Log[x] + (60*x^2 + 76*x^3 + 31*x^4 + 4*x^5)*Log[x]*Log[Log[x]])/((4 + 4*x + x^2)*Log[x]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {20 x^2+24 x^3+9 x^4+x^5+\left (12+20 x+221 x^2+208 x^3+71 x^4+8 x^5\right ) \log (x)+\left (60 x^2+76 x^3+31 x^4+4 x^5\right ) \log (x) \log (\log (x))}{\left (4+4 x+x^2\right ) \log (x)} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ 2 x^4+x^4 \log (\log (x))+13 x^3+5 x^3 \log (\log (x))+10 x^2-15 x-\frac {72}{x+2} \]