As $M \to +\infty$, how could I make a good asymptotic analysis of this integral?

$$\int_0^1 \dfrac{\cos(M x)}{1 + x^2} e^{-M (x^2 - 1/9)}\ \text{d}x$$

The exponential term shall dominate, yet I have no clue in who to deal with $\cos(Mx)$. I tried to apply geometric series for $\frac{1}{1+x^2}$ but an infinite sums might not be the best deal.

I am not sure if/when to use Taylor series. I thought about Laplace Method, but I think it could not work because of the $\cos(MX)$ function... Any hint?

Thank you!