$\int_0^\infty\frac{\cos(ax)}{x^4+5x^2+4}\,dx$ for $a\in\mathbb{R}$

Factor \[ x^4+5x^2+4=(x^2+1)(x^2+4) \] and decompose \[ \frac1{(x^2+1)(x^2+4)} =\frac13\left(\frac1{x^2+1}-\frac1{x^2+4}\right). \] For \(b>0\) and real \(a\), the standard upper-half-plane residue computation gives \[ \int_0^\infty\frac{\cos(ax)}{x^2+b^2}\,dx =\frac{\pi}{2b}e^{-b|a|}. \] Therefore \[ \begin{aligned} \int_0^\infty\frac{\cos(ax)}{x^4+5x^2+4}\,dx &=\frac13\left(\frac\pi2e^{-|a|}-\frac\pi4e^{-2|a|}\right)\\ &=\boxed{\frac\pi{12}\left(2e^{-|a|}-e^{-2|a|}\right)}. \end{aligned} \]