$$\frac{1}{\pi}\int_{0}^\infty \exp(-\gamma q^\alpha)\cos(qx)\,dq$$
$$=\frac{1}{\pi}\gamma^{-1/\alpha} \Gamma \left(1+\frac{1}{\alpha}\right)-\frac{x^2 \gamma^{-3/\alpha} \Gamma \left(\frac{3}{\alpha}\right)}{2\pi \alpha}+{\cal O}(x^4).$$