Yesterday I wrote some simple integrals involving nested radicals, and/or continued fractions. This was an example with nested radicals, let $$\int_0^1\frac{dx}{\sqrt{1+(\log x)g(x)}},\tag{1}$$ where $g(x)=\sqrt{1+(\log(x)\cdot g(x))}$, that is $$g(x)=\sqrt{1+\log x^{g(x)}}.\tag{2}$$
Question. Calculate a good approximation of previous integral. Thanks in advance.
You approach can be using analysis or a numerical method.
One has that $(g(x))^2=1+g(x)(\log x)$ and thus we need to calculate, if there are no mistakes, $$\sqrt{2}\int_0^1\frac{dx}{\sqrt{2+\log^2(x)+\sqrt{\log ^2(x)+4}}}.$$