Question

Is there any exact formula or at least exact inequalities for the following intehral

$$ \int_2^x\frac{dt}{[\frac{\log x}{\log t}]\log t} $$

where [x] is the greatest integer less than or equal to x.

added:

When I use

$$ x-1<[x]\le x $$ I get $$ \frac{x-2}{\log x}=\int_2^x\frac{dt}{\log x}\leq \int_2^x\frac{dt}{[\frac{\log x}{\log t}]\log t}\le \int_2^x\frac{dt}{\log x-\log t} $$ but they are not exact enough. I need more closer bounds.