integration – Asymptotic expansion / analysis of this integral


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!