I need help finding the analytical solution of this integral, if any. Any insights are much appreciated. A lower/upper bound is also good.
$$\int_{-\infty}^{\infty}\!\frac{e^{\frac{1}{2}(t+2\sqrt{t}x-x^2)}}{\sqrt{2\pi}(1+(e^{\frac{t}{2}+\sqrt{t}x}-1)q)}\,\mathrm{d}x$$ for $q\in[0,1], t\geq0$.
I know that when $q=1$ or $t=0$ it evaluates to $1$, and to $e^t$ when $q=0$.
For background, this integral is part of a utility function as part of a research I'm doing, where $q$ is the person's belief, and hence a probability between $[0,1]$, $t$ is time and $x$ is the dummy variable of integration. It is derived using the normal distribution hence the exponential and $\sqrt{2\pi}$ terms.