So many logs and exponents

Calculus Level 4

\[ \Large J=\int_{^{n+1}e}^{^{n+2}e} \dfrac {dx}{ x \prod_{i=1}^n \underbrace{\ln ( \ln ( \ln \cdots ( \ln x ) ) \cdots )}_{i \text{ times} }} \]

Let \(^A B \) denote the tetration function, \(^A B = \underbrace{B^{B^{B^{\cdot^{\cdot^{\cdot^B}}}}}}_{ \text{A times} } \).

Find \( \lfloor J + 2 \rfloor \).

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