\[ \int \frac {-x^2+2 x \log (25)-\log ^2(25)+\left (15 x^2-20 x \log (25)+5 \log ^2(25)\right ) \log (x)+\left (-3 x^2+4 x \log (25)-\log ^2(25)\right ) \log (x) \log (\log (x))}{\left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)\right ) \log (x)+\left (x^3-2 x^2 \log (25)+x \log ^2(25)\right ) \log (x) \log (\log (x))} \, dx \]
Optimal antiderivative \[ -\ln \left (\left (x -2 \ln \left (5\right )\right )^{2} x \left (5-\ln \left (\ln \left (x \right )\right )\right )+2\right ) \]
command
Int[(-x^2 + 2*x*Log[25] - Log[25]^2 + (15*x^2 - 20*x*Log[25] + 5*Log[25]^2)*Log[x] + (-3*x^2 + 4*x*Log[25] - Log[25]^2)*Log[x]*Log[Log[x]])/((-2 - 5*x^3 + 10*x^2*Log[25] - 5*x*Log[25]^2)*Log[x] + (x^3 - 2*x^2*Log[25] + x*Log[25]^2)*Log[x]*Log[Log[x]]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ -\log \left (5 x^3+x^3 (-\log (\log (x)))+2 x^2 \log (25) \log (\log (x))-10 x^2 \log (25)-x \log ^2(25) \log (\log (x))+5 x \log ^2(25)+2\right ) \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {-x^2+2 x \log (25)-\log ^2(25)+\left (15 x^2-20 x \log (25)+5 \log ^2(25)\right ) \log (x)+\left (-3 x^2+4 x \log (25)-\log ^2(25)\right ) \log (x) \log (\log (x))}{\left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)\right ) \log (x)+\left (x^3-2 x^2 \log (25)+x \log ^2(25)\right ) \log (x) \log (\log (x))} \, dx \]________________________________________________________________________________________