So many logs and exponents

$L_{i}(x) = \underbrace{\ln ( \ln ( \ln \cdots ( \ln x ) ))}_{i \text{ times}}$
$J = \displaystyle \int_{ ^{n+1}e}^{ ^{n+2}e} \dfrac{1}{xL_{1}(x)L_{2}(x)\ldots L_{n}(x)} dx$
$L_{n}(x) = t$
$\dfrac{1}{xL_{1}(x)L_{2}(x) \ldots L_{n-1}(x)}dx = dt$
$\therefore J =\displaystyle \int_{e}^{e^{e}} \dfrac{dt}{t} = \left[ \ln(t) \right]_{e}^{e^{e}} = e - 1$
$\lfloor J + 2 \rfloor = \lfloor e - 1 + 2 \rfloor = \lfloor e + 1 \rfloor$

$2 < e < 3$
$3 < e+ 1 < 4$
$J = \lfloor e + 1 \rfloor = 3$

It is greatest integer of j not j!