How to Evaluate $\int_{0}^{\infty}\frac{x^2+2}{x^4+4} \ dx$ given $\mathcal{F}_c(e^{-x}\cos(x))=\sqrt{\frac{2}{\pi}}\frac{k^2+2}{k^4+4}$

Question

I have previously shown that ($\mathcal{F}_c(f(x))$ denotes the Fourier cosine transform of $f(x)$) $$\mathcal{F}_c(e^{-x}\cos(x))=\sqrt{\frac{2}{\pi}}\frac{k^2+2}{k^4+4} \tag{1}.$$ Using this information, I am trying to solve the integral $$\int_{0}^{\infty}\frac{x^2+2}{x^4+4} \ dx.$$ I thought the problem could be solved by taking the inverse Fourier cosine transform of $(1)$: $$e^{-x}\cos(x)=\mathcal{F}_c^{-1}\left(\sqrt{\frac{2}{\pi}}\frac{k^2+2}{k^4+4}\right).$$ But clearly this leads to $$e^{-x}\cos(x)=\frac{2}{\pi}\int_0^{\infty}\frac{k^2+2}{k^4+4}\cos(kx) \ dk.$$ I do not require a full solution at this stage, just a hint on how to proceed. I wish to solve this homework problem as much as I can.