This Many Logs Inside An Integral Makes Me Sick!

Calculus Level 4

\[ \large \int_e^{e^{512}} \dfrac{dx}{x \ln x + x \ln x ( \ln (\ln x))^2 } \]

If the integral above is equal to \( \arctan\left(\ln A^{B^C} \right) \), where \(A,B\) and \(C\) are prime numbers, find \(A+B+C\).

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