Examine for conditional and absolute convergence: $\int_1^\infty \frac{\cos y\,\mathrm{d}y}{(y+\sin y)^\alpha}$

Question

I need to examine the integral $$\int\limits_1^\infty \frac{\cos y \,\mathrm{d}y}{(y+\sin y)^\alpha}$$ for conditional and absolute convergence based on $\alpha$'s values. Since this integral looks like none of the templates we use, I tried going the following way: $$\int\limits_1^\infty\left|\frac{\cos y\,\mathrm{d}y}{(y+\sin y)^\alpha}\right|\le\int\limits_1^\infty \cfrac{\mathrm{d}y}{(y+\sin y)^\alpha}$$ but then I don't know what to do. The only way of going forward that I can think of is substitution but I don't think it will be of any use. Note that the problem was originally with $\frac{1}{x}$ instead of $y$ but I thought it would be a good idea to simplify it. Could someone give me at least a hint, please?