Solving a problem with one dimensional diffusion the following identity naturally arises
$$\int _{-\infty }^{\infty }\frac{q}{2}{{\rm e}^{\eta\,t{\alpha}^{2}-\alpha \,x}} \left( {{\rm erf}\left({\frac {-2\,\alpha\,\eta\,t+x+a}{2\sqrt {\eta t}}}\right)} + {{\rm erf}\left({\frac {2\,\alpha\,\eta\,t-x+a}{2\sqrt {\eta t}}}\right)} \right) {dx}=-{\frac {q \left( -{{\rm e}^{\alpha\,a}}+{{\rm e}^{- \alpha\,a}} \right) }{\alpha}} $$
Please let me know how to prove such identity directly.