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Calculus Level 5

$\large I=\int _0^1 \int _0^x \frac {x^4}{\ln(s) \sqrt {\ln\left(\frac { 1 }{ x } \right)}} \ ds\ dx$ 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 $$\overline{MAA}$$.

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