Following the approach in this question, for $$\phi (t) = \frac{A} {1 +\exp\Big(B (C+ D t) \Big)} $$
I derived
$$u = \frac{2}{\sqrt \pi}\int_{\mu=0}^{\infty} \frac{A} {1 +\exp\Big(B (C+ D (t - \frac{x^2}{4\kappa\mu^2})) \Big)} e^{-\mu^2} d\mu \\ - \frac{2}{\sqrt \pi}\int_{\mu=_0}^{x/2\sqrt{4\kappa t}} \frac{A} {1 +\exp\Big(B(C+ D (t - \frac{x^2}{4\kappa \mu^2})) \Big)} e^{-\mu^2} d\mu$$
However, I cannot make the integration to proceed further.