$\large \int_{\frac{1}{8}}^{\frac{1}{2}} \left \lfloor \ln \left \lceil \dfrac{1}{x} \right \rceil \right \rfloor dx = \, ?$

$$ **Notations**: $\lfloor \cdot \rfloor$ denotes the floor function and $\lceil \cdot \rceil$ denotes the ceiling function.

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