I have a random variable $X \sim \operatorname{Gamma}(\alpha, \beta)$.

How can I compute or approximate $\mathbb{E} \sin(X)$ very quickly? Iterative quadrature would be too slow, I need some closed form expression.

One idea I considered was to use the Gamma moments and Tyler approximation, but it would take too many terms, since $X$ has large standard deviation (in the order of 10–50) and so the mass is not tightly concentrated.