I would like to sum the series 
$$
\sum\_{n=0}^\infty \frac{1}{(1+a^2 (n+1/2)^2) ^{3/2}} .
$$
It arose when trying to perform a calculation on superconductivity. In particular I am interested in its diverging behaviour for $a\to 0$.

I have tried turning it into a contour integral by defining
$$
f(z)=tan(\pi z)(1+a^2 z^2) ^{-3/2}
$$

The integral along the real axis gives (modulo factors of $\pi$ and 2) the desired sum. The integral over a great cicle in the upper half plane should vanish, but there is a branch cut to take care of. When I choose it to lie from $\frac i a$ to $+i \infty$ my integral along this branch cut from $i \infty$ to the pole diverges and I can get no meaningful answer. Maybe I am using the wrong technique for evaluating this sum. Mathematica also refuses to solve it.

Any suggestion on this problem would be much appreciated.