What I've done so far:
$$\dfrac{1}{2\sqrt{2\pi}}\left[\int_{-\infty}^{\infty}\dfrac{e^{i(\beta + k)x}}{a^{4} + x^{4}}dx +\int_{-\infty}^{\infty}\dfrac{e^{i(k - \beta)x}}{a^{4} + x^{4}}dx\right] = \\\int_{-\infty}^{\infty}\dfrac{e^{i(\beta + k)x}}{(a^2 + ix^2)(a^{2} -ix^2)}dx +\int_{-\infty}^{\infty}\dfrac{e^{i(k - \beta)x}}{(a^2 + ix^2)(a^{2} -ix^2)}dx]$$
considering that de Fourier transform is given by:
$$F(K) = \dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{ikx}dx$$
My problem is that I can't find a way of plotting the poles of the function so I can find a integration contourn