Integral of the Floor Function of a Logarithm

Write \lceil x\rceil for the least integer greater than x and \lfloor x \rfloor for the greatest integer less than x. Eg. \lceil 1.5\rceil = 2 and \lfloor 1.5 \rfloor = 1.

Evaluate:

\displaystyle \int_0^\infty \left\lfloor \log_a \left\lfloor \frac {\lceil x \rceil} x \right\rfloor \right\rfloor \mathrm dx

for a > 0.

[originally posted by Kummer on TSR]

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