I am looking for an explanation of the bound $$\frac{1}{2\pi}\left(-\frac{T \log T}{1+(t-T)^2} - 2 \int_T^\infty \frac{x \log x (t-x)}{(1+(t-x)^2)^2} dx \right)\ll \left( \frac{1}{t+1} + \frac{1}{T-t+1}\right) \log T,$$
where $$t \in [0,T].$$
This is not for homework. Instead, this is a question from my senior thesis reading, so full answers are appreciated. I can bound the first term by $$\frac{\log T}{T - t +1}$$, but I am having trouble bounding the second.