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.

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.

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# integrate of functions involving floor

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