Evaluate $\int_0^\infty e^{-x^2}\cos\frac{t}{x^2}\,\mathrm dx$ How to integrate 

$$\int_0^\infty e^{-x^2}\cos\frac{t}{x^2}\,\mathrm dx$$
where $t$ is a constant/parameter?
Maybe:
$$\int_0^\infty e^{-x^2}\cos\frac{t}{x^2}dx=\int_0^\infty \frac{1}{2}e^{-x^2}\left(e^\frac{it}{x^2}+e^\frac{-it}{x^2}\right) dx\\
=\int_0^\infty \frac{1}{2}\left(e^{-x^2+\frac{it}{x^2}}+e^{-x^2-\frac{it}{x^2}}\right) dx$$

Complete the square for the second term $\left(x+\sqrt t\frac{1-i}{\sqrt2}x^{-1}\right)^2-\sqrt {2t}i=x^2+itx^{-2}$

And similar for the first term... 

Is it possible to integrate this out with the substitutions $$u=x+(1-i)\sqrt\frac{t}{2}x^{-1}$$ and 
$$s=x+(1+i)\sqrt\frac{t}{2}x^{-1}$$?
I can give you only the results for different $t$s.

Let $t=1$ be, then 

$$\int _0^{\infty }e^{-x^2}\cos \left(\frac{1}{x^2}\right)dx = \frac{e^{-\sqrt{2}}\sqrt{\pi}\cos(\sqrt{2})}{2}$$

If $t=2$ is, then
$$\int _0^{\infty }e^{-x^2}\cos \left(\frac{2}{x^2}\right)dx = \frac{e^{-2}\sqrt{\pi}\cos(\sqrt{4})}{2}$$
If $t=3$, then
$$\int _0^{\infty }e^{-x^2}\cos \left(\frac{3}{x^2}\right)dx = \frac{e^{-\sqrt{6}}\sqrt{\pi}\cos(\sqrt{6})}{2}$$
So:
Is $t$ a constant, then we have:
$$\int _0^{\infty }e^{-x^2}\cos \left(\frac{t}{x^2}\right)dx = \frac{e^{-\sqrt{2t}}\sqrt{\pi}\cos(\sqrt{2t})}{2}$$