\[ \int \frac {-25+20 x^2+\left (5 x-4 x^3\right ) \log (5)+\left (20 x^2-4 x^3 \log (5)\right ) \log ^2(x)+(-5+x \log (5)) \log \left (\frac {5-x \log (5)}{\log (5)}\right )+\log (x) \left (-25-20 x^2+\left (4 x+4 x^3\right ) \log (5)+(-5+x \log (5)) \log \left (\frac {5-x \log (5)}{\log (5)}\right )\right )}{\left (60 x^2-20 e^3 x^2+20 x^3+\left (-12 x^3+4 e^3 x^3-4 x^4\right ) \log (5)\right ) \log ^2(x)+\log (x) \left (25 x-20 x^3+\left (-5 x^2+4 x^4\right ) \log (5)+\left (5 x-x^2 \log (5)\right ) \log \left (\frac {5-x \log (5)}{\log (5)}\right )\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left ({\mathrm e}^{3}+\frac {x -\frac {5+\ln \left (\frac {5}{\ln \left (5\right )}-x \right )}{4 x}}{\ln \! \left (x \right )}-3-x \right ) \]
command
Int[(-25 + 20*x^2 + (5*x - 4*x^3)*Log[5] + (20*x^2 - 4*x^3*Log[5])*Log[x]^2 + (-5 + x*Log[5])*Log[(5 - x*Log[5])/Log[5]] + Log[x]*(-25 - 20*x^2 + (4*x + 4*x^3)*Log[5] + (-5 + x*Log[5])*Log[(5 - x*Log[5])/Log[5]]))/((60*x^2 - 20*E^3*x^2 + 20*x^3 + (-12*x^3 + 4*E^3*x^3 - 4*x^4)*Log[5])*Log[x]^2 + Log[x]*(25*x - 20*x^3 + (-5*x^2 + 4*x^4)*Log[5] + (5*x - x^2*Log[5])*Log[(5 - x*Log[5])/Log[5]])),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {-25+20 x^2+\left (5 x-4 x^3\right ) \log (5)+\left (20 x^2-4 x^3 \log (5)\right ) \log ^2(x)+(-5+x \log (5)) \log \left (\frac {5-x \log (5)}{\log (5)}\right )+\log (x) \left (-25-20 x^2+\left (4 x+4 x^3\right ) \log (5)+(-5+x \log (5)) \log \left (\frac {5-x \log (5)}{\log (5)}\right )\right )}{\left (60 x^2-20 e^3 x^2+20 x^3+\left (-12 x^3+4 e^3 x^3-4 x^4\right ) \log (5)\right ) \log ^2(x)+\log (x) \left (25 x-20 x^3+\left (-5 x^2+4 x^4\right ) \log (5)+\left (5 x-x^2 \log (5)\right ) \log \left (\frac {5-x \log (5)}{\log (5)}\right )\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \log \left (-4 x^2+4 x^2 \log (x)+4 \left (3-e^3\right ) x \log (x)+\log \left (\frac {5}{\log (5)}-x\right )+5\right )-\log (x)-\log (\log (x)) \]