I want to evaluate the sums
$$ \sum_{q=-\infty}^\infty\frac{1}{\sin\left(a q + x\right)\left(q + b \right)^4} $$ and $$ \sum_{q=-\infty}^\infty\frac{\cos\left(a q + x\right)}{\sin^2\left(a q + x\right)\left(q + b \right)^3} $$
where $x\in\mathbb{C}$, and $a,b\in\mathbb{R}$.
Do either of these sums have a closed form answer? I suspect not. They appear as part of an expression that I'm attempting to numerically solve (in Mathematica) for $x$. I know that I can just approximate each sum with many terms, but numerically solving this is very slow. I'm hoping that maybe one or both of these sums can be represented in terms of, say, some of Mathematica's built in functions. Any ideas?