Evaluating $\int_0^\infty \frac{\cos(ax)-e^{-ax}}{x \left(x^4+b^4 \right)}dx$

Evaluating $\int_0^\infty \frac{\cos(ax)-e^{-ax}}{x \left(x^4+b^4 \right)}dx$

How can we evaluate
$$\int_0^\infty \frac{\cos(ax)-e^{-ax}}{x \left(x^4+b^4\right)}dx \quad
a,b>0$$
using Complex Analysis? This problem was given in a Complex Analysis book
which I was reading. The answer given in it is
$$\frac{\pi}{4b^4}e^{-ab}\sin \left(\frac{ab}{\sqrt{2}} \right)$$
Which function and contour should we consider?