\[ \int \frac {2+x-2 x^2-x \log (16)+\left (-x+x^2+(-1+x) \log (16)-\log \left (x^2\right )\right ) \log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right ) \log \left (\log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right )\right )+\left (-x+x^2+(-1+x) \log (16)-\log \left (x^2\right )\right ) \log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right ) \log \left (\log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right )\right ) \log \left (\log \left (\log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right )\right )\right )}{\left (x^3-x^4+\left (x^2-x^3\right ) \log (16)+x^2 \log \left (x^2\right )\right ) \log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right ) \log \left (\log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right )\right )} \, dx \]
Optimal antiderivative \[ \frac {\ln \! \left (\ln \! \left (\ln \! \left (\ln \! \left (x^{2}\right )+\left (x +4 \ln \! \left (2\right )\right ) \left (1-x \right )\right )\right )\right )+1}{x} \]
command
Integrate[(2 + x - 2*x^2 - x*Log[16] + (-x + x^2 + (-1 + x)*Log[16] - Log[x^2])*Log[x - x^2 + (1 - x)*Log[16] + Log[x^2]]*Log[Log[x - x^2 + (1 - x)*Log[16] + Log[x^2]]] + (-x + x^2 + (-1 + x)*Log[16] - Log[x^2])*Log[x - x^2 + (1 - x)*Log[16] + Log[x^2]]*Log[Log[x - x^2 + (1 - x)*Log[16] + Log[x^2]]]*Log[Log[Log[x - x^2 + (1 - x)*Log[16] + Log[x^2]]]])/((x^3 - x^4 + (x^2 - x^3)*Log[16] + x^2*Log[x^2])*Log[x - x^2 + (1 - x)*Log[16] + Log[x^2]]*Log[Log[x - x^2 + (1 - x)*Log[16] + Log[x^2]]]),x]
Mathematica 13.1 output
\[ \int \frac {2+x-2 x^2-x \log (16)+\left (-x+x^2+(-1+x) \log (16)-\log \left (x^2\right )\right ) \log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right ) \log \left (\log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right )\right )+\left (-x+x^2+(-1+x) \log (16)-\log \left (x^2\right )\right ) \log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right ) \log \left (\log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right )\right ) \log \left (\log \left (\log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right )\right )\right )}{\left (x^3-x^4+\left (x^2-x^3\right ) \log (16)+x^2 \log \left (x^2\right )\right ) \log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right ) \log \left (\log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right )\right )} \, dx \]
Mathematica 12.3 output
\[ \frac {1}{x}+\frac {\log \left (\log \left (\log \left (-((-1+x) (x+\log (16)))+\log \left (x^2\right )\right )\right )\right )}{x} \]