Summation using residues

Here is one way to consider the series.

Expand it as

$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n(e^{\pi n}+1)}-\frac{1}{2}\sum_{n=1}^{\infty}\frac{1}{n(e^{2\pi n}+1)}+\sum_{n=1}^{\infty}\frac{1}{n(e^{\pi n}-1)}-\frac{1}{2}\sum_{n=1}^{\infty}\frac{1}{n(e^{2\pi n}-1)}=\frac{log(2)}{8}$

These are rather famous sums studied by Ramanujan. We can probably find their evaluations in his notebooks or some other place like Plouffe, as was previously mentioned.

Can any of the above sums be evaluated using residues?.

I think there may be a general formula derived by Ramanujan for sums of this type.

I think I have seen one in terms of Bernoulli numbers.

Anyway, some thoughts.

Fulfilling joriki's request, I am providing this "no-go" answer.

The method OP wished to apply rests on the assumption (that I did not verify) that the integral around the big square $C_N$ goes to zero. If this is the case, the sum of the residues is also zero and therefore this sum can be split into two equal parts -- the one coming from $\cot(\pi z)$ representing the original sum and the part coming from the poles of the function $f(z)$ itself.

If $f(z)$ had only few singularities, we would succeed in solving the problem. This is not the case here though, since $\sinh(\pi(2z + 1)) = (-i)\sin(i\pi(2z + 1))$ has infinitely many zeros on the line $-1/2 + it$, so the resulting sum is about as complicated as the original sum. Therefore it would seem this method is not applicable for this problem.