\[ \int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx \]
Optimal antiderivative \[ \frac {\left (5+\frac {x -\ln \left (\frac {5}{2 x \ln \left (x \right )^{2}}\right )}{\ln \left (3\right )}\right )^{2}}{x -\frac {4}{x}} \]
command
Integrate[(-16*x + 4*x^3 + (-80 + 20*x^2)*Log[3] + (-8*x - 12*x^2 + 2*x^3 + x^4 + (-40 - 80*x + 10*x^2)*Log[3] + (-100 - 25*x^2)*Log[3]^2)*Log[x] + (16 - 4*x^2 + (8 + 16*x - 2*x^2 + (40 + 10*x^2)*Log[3])*Log[x])*Log[5/(2*x*Log[x]^2)] + (-4 - x^2)*Log[x]*Log[5/(2*x*Log[x]^2)]^2)/((16 - 8*x^2 + x^4)*Log[3]^2*Log[x]),x]
Mathematica 13.1 output
\[ \frac {\int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log (x)} \, dx}{\log ^2(3)} \]
Mathematica 12.3 output
\[ \int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx \]________________________________________________________________________________________