integral of exponential (Green) function with trigonometric variable

Question

I am searching for a tip on how to handle the following integral, $\int_c^\infty e^{-x^2}\sin\dfrac{b}{x^2}~dx$ and also $\int_c^\infty e^{-x^2}\cos\dfrac{b}{x^2}~dx$ where both $c$ and $b$ are real. I could not find it in the Integrals table. Thanks for having a look!