For example, two Green functions:
\begin{equation} G_1(\tau_1 - \tau_2, x) = \alpha_1(\tau_1 - \tau_2)\beta_1(x) H(\tau_1 - \tau_2) + \mu_1(\tau_1 - \tau_2) \nu_1 (x) H(\tau_2 - \tau_1) \end{equation}
\begin{equation} G_2(\tau_2 - \tau_3, x') = \alpha_2(\tau_2 - \tau_3)\beta_2(x') H(\tau_2 - \tau_3) + \mu_2(\tau_2 - \tau_3) \nu_2 (x') H(\tau_3 - \tau_2) \end{equation}
where the H is a step function, $H(\tau_a - \tau_b)=1$ when $\tau_a > \tau_b$ and zero otherwise. Can a product of these two Green functions be integrated? -
\begin{equation} \int_0^y d\tau_2 G_1(\tau_1 - \tau_2, x) G_2(\tau_2 - \tau_3, x') \end{equation}
If they can be, is there a general method to do so?
Edit: For an easier comment below, I'll use
\begin{equation} G_1(\tau_1 - \tau_2, x) = c H(\tau_1 - \tau_2) + d H(\tau_2 - \tau_1) \end{equation}
\begin{equation} G_2(\tau_2 - \tau_3, x') = a H(\tau_2 - \tau_3) + b H(\tau_3 - \tau_2) \end{equation}