Analytical approximation for an integral $\int_{-\infty}^\infty\frac{dx}{x^4 + a^2t^4 - 2at^2x^2\cos(x) + a^2t^2x^2 - 2atx^3\sin(x)}$

Question

I am looking for an approximation/solution to this integral:

$$\int_{-\infty}^{\infty}\frac{dx}{x^4 + a^2t^4 - 2at^2x^2\cos(x) + a^2t^2x^2 - 2atx^3\sin(x)}.$$

Particularly, I'm looking for the case that $a>0$ and $t>0$.