I am trying to calculate the residues of $\frac{e^{imz}}{1+z^4}$. For context, I am calculating $\int\limits_0^\infty \frac{e^{imz}}{1+z^4}dx$ and taking the real part of the integral to calculate $\int\limits_0^\infty \frac{\cos mx}{1+x^4}dx$. I don't necessarily want the answer to the final integral, I would prefer if someone helped me with computing the residues since they seem overly complicated.
We have that $f(z) =\frac{e^{imz}}{1+z^4}$ has simple poles at $e^{\frac{\pi i}{4}}, e^{-\frac{\pi i}{4}}, e^{\frac{3\pi i}{4}},$ and $e^{-\frac{3\pi i}{4}}$. Based on my choice of contour, the only poles I need to calculate the residue for are $z=e^{\frac{\pi i}{4}}$ and $z=e^{\frac{3\pi i}{4}}$. We have that \begin{align*} Res(f, e^{\frac{\pi i}{4}}) &= \frac{exp(ime^{\frac{\pi i}{4}})}{4e^{\frac{3\pi i}{4}}}\\ Res(f, e^{\frac{3\pi i}{4}}) &= \frac{exp(ime^{\frac{3\pi i}{4}})}{4e^{\frac{\pi i}{4}}} \end{align*} When I try to apply residue theorem to the desired integral, I get a very complicated expression, and so I am not too sure if the residues I computed are correct. Did I compute the residues correctly?