How to bound this integral?

I want to ask how a hint how to show this integral inequality: $$ \frac{1}{\pi}\int^{\infty}_{0}\frac{x}{y^{2}+x^{2}}\log\frac{1}{1-e^{-2\pi y}}dy< \frac{1}{12x} $$ This is from Ahlfors, Complex Analysis, page $206$. I tried to compute a rough bound, but my bound is too rough and it did not work.

Explicit computation showed the bound is quite delicate, for example for $x=6000$ the value on the left hand side is $0.000013888888876028806686426283065381014133382406390603...$ as opposed to $0.00001388888888...$. For $x=60000$ the accuracy is about $0.99999999999074074074$, for $x=75000$ the accuracy is about $0.99999999999407407407$.

This is the refinement term in the usual Stirling's formula. I know it for a long time but I never knew how to prove it.

Solution 1:

Hint 1:

$${x^2\over x^2+y^2}\lt1$$

Hint 2:

$${1\over w}\log{1\over1-w}=1+{1\over2}w+{1\over3}w^2+\cdots$$

(Note: I'll flesh this out if the OP requests, but all the questions asked for was a hint.)