I'm working on a numerical simulation of population dynamics that would be greatly sped up if I could solve the following definite integral:
$$\int_{0}^te^{-[x-\theta(T+\tau)]^2/2\sigma^2}d\tau $$
where $\theta(x)$ is any well behaved sigmoid function such as $\tanh(x)$. All the parameters are real-valued and $\sigma>0$. Thanks in advance for any ideas!