Consider the logarithmic integral $\operatorname{Li}(x):=\int_2^x \frac{dt}{\log t}.$
Then I found a result stating that we have $\operatorname{Li}(x)=x/\log x+O(x/\log^2(x))$ and another integration by parts gives $\operatorname{Li}(x)=x/\log x + x/\log^2 (x)+O(x/\log^3 (x))$.
I can see that integration by parts gives $\int_2^x dt/ \log t=x / \log x - 2/ \log x +\int_2^x dt/ log^2(t)$.
And integrating by parts again gives $\int_2^x dt/ \log t=x / \log x - 2/ \log x +x/\log^2 x-2/\log^2 2 + 2\int_2^x dt/\log^3t.$
So it seems like we should have $\int_2^x dt/\log^3 t = O(x/\log^3 x)$ but I don't know how to prove this. I would greatly appreciate any help.