On the evaluation of $\int_0^\infty e^{-x^2} \sin \left (\frac{1}{4x^2} \right ) \, dx$

Following tired's suggestion, the problem is equivalent to finding

$$\text{Im}\int_{0}^{+\infty}\exp\left(-x^2+\frac{i}{4x^2}\right)\,dx =\frac{1}{2}\text{Im}\int_{-\infty}^{+\infty}\exp\left(-\left(x-\frac{1-i}{2\sqrt{2}\,x}\right)^2-\frac{1-i}{\sqrt{2}}\right)\,dx$$ and through a rotation of the integration line this can be easily computed through the Cauchy-Schlömilch substitution, the best known instance of Glasser's master theorem: $$\forall a\in\mathbb{R}^+,\qquad \int_{0}^{+\infty}\exp\left(-x^2-\frac{a}{x^2}\right)\,dx = \frac{\sqrt{\pi}}{2}\,e^{-2\sqrt{a}}.$$