I am new to math.stackexchange.com. I am trying to develop a fitting routine and I encounter an integral as follows: $$\int_{0}^{x}\mathrm{erf\left(\frac{t-a}{m}\right)} \cdot t \cdot J_{0}(qt)\mathrm{d}t $$ I have an analytic solution in the following form: $$ \int_{0}^{x} \mathrm{erf(t)}\cdot t \cdot J_{0}(qt) \mathrm{d}t $$
Thus, I want your help to express the $\mathrm{erf(\frac{t-a}{m})}$ in form of sum of $\mathrm{erf(x)}$. I can try to handle any extra variables(t) that may emerge during this transformation.
Any help would be greatly appreciated.