You might try to regularize the sum,
$$\sum_{n=1}^\infty \frac{ \cos(\alpha n) \sin(n+1) }{n}=
-\frac{1}{4} i e^{-i} \left(\ln \left(1-e^{i (\alpha-1)}\right)+\ln \left(1-e^{-i (\alpha+1)}\right)-e^{2 i} \left(\ln \left(1-e^{-i (\alpha-1)}\right)+\ln  \left(1-e^{i (\alpha+1)}\right)\right)\right).$$
This is finite for $\alpha\neq 1$.