Evaluate contour integral

Question

I have a homework problem to evaluate the integral
$$
\oint_{\gamma}\frac{\cos z}{(z+i)^3}dz
$$
along the curve $\gamma(t)=-i+e^{it}, t\in[0,2\pi]$. I proceeded to plug the given information into the definition of a contour integral and got to the expression
$$
\oint_{\gamma}=i\int_{0}^{2\pi}\cos(-i+e^{it})e^{-2it}dt
$$
which seems hardly helpful. I don't know how to evaluate this or manipulate it any further and I suppose there is some trick earlier on to make the integration more manageable. I just can't figure it out so I'd be grateful for any help.

Answer 1

With rediue theorem $$\oint_{\gamma}\frac{\cos z}{(z+i)^3}dz=\dfrac{2\pi i}{2!}\lim_{z\to-i}\dfrac{d^2}{dz^2}\cos z=-\dfrac{2\pi i}{2!}\cos i=\color{blue}{-\dfrac{\pi i}{2}\left(e+\dfrac1e\right)}$$