I would like to compute the following function: $$f(t)=\mathcal{L}^{-1}\Big[\frac{1}{s(e^{a+\text{arcosh}(s+\cosh a)}-1)}\Big](t)$$
However, it seems that there is no other pole than the pole of order one $s=0$, which gives a residue without $t$-dependence. I guess that I should consider that $\text{arcosh}(\cosh a)=\pm a$, in which case it seems (always for the negative case) that the pole is essential.
Why should I consider that $\text{arcosh}$ is negative? How can I compute this residue anyway?