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$\large I=\int _0^1 \int _0^x \frac {x^4}{\ln(s) \sqrt {\ln\left(\frac { 1 }{ x } \right)}} \ ds\ dx$ If $I$ can be expressed as $\displaystyle -2\sqrt { \frac \pi A} \ln(\sqrt A +\sqrt M)$ , where $A$ and $M$ are square-free, find the 3-digit integer $\overline{MAA}$ .