This is what I tried:
$$\sum_{k=0}^{\infty}\int_{-\infty}^{\infty}\dfrac{(x-n\sinh^2 x)^k}{k!}\ dx$$
$$\sum_{k=0}^{\infty}\frac1{k!}\int_{-\infty}^{\infty}\sum_{r=0}^{k}\binom{k}{r}x^r(-n\sinh^2 x)^{k-r}\ dx$$
After this I have no idea how to proceed