I am trying to get a closed form analytic result for the integral $$\int _{0}^{\infty }\!{\frac {\left(1-{{\rm e}^{-i \left( {q}-{p} \right) t}}\right){\rm ln}(|p^2-p_0^2|)}{ ( {q}-{p} ) \left( {{ p}}^{2}-{{p_1}}^{2} \right) \left( {{p}}^{2}-{{p_2} }^{2} \right) }}{d{p}} $$ $p_1$, $p_2$ $\in \mathbb{C}$ with $Im(p_1),Im(p_2)<0$ and $Re(p_1)<0$, $Re(p_2)>0$, $q_0,q,t>0$ and $p_0\not =q$. The numerical values I obtain are finite. The integrand has an integrable singularity at $p=p_0$ due to the ${\rm ln}$, but no singularity at $p=q$ due to the presence of the numerator. It is not obvious to me that it may be represented in closed form, but I had some success evaluating the simplified integral $$\int _{0}^{\infty }\!{\frac {1-{{\rm e}^{-i \left( {\it q}-{\it p} \right) t}}}{ ( {\it q}-{\it p} ) \left( {{ \it p}}^{2}-{{\it p_1}}^{2} \right) \left( {{\it p}}^{2}-{{\it p_2} }^{2} \right) }}{d{\it p}} $$ Mathematica and Maple return closed form results with exponential integral functions for it if one introduces $+i\epsilon$ in the denom, $$\int _{0}^{\infty }\!{\frac {1-{{\rm e}^{-i \left( {\it q}-{\it p} \right) t}}}{ ( {\it q}-{\it p} +i\epsilon) \left( {{ \it p}}^{2}-{{\it p_1}}^{2} \right) \left( {{\it p}}^{2}-{{\it p_2} }^{2} \right) }}{d{\it p}} $$ and specifies the assumptions on the variables properly. $\epsilon$ can be sent to zero after the integration.
It is also possible to evaluate the original integral if the exponential is dropped and a small epsilon is introduced.
For the original integral I tried various things like integration by parts with respect to the logarithm, expansion of the logarithm, complex decomposition. Any ideas? It would maybe also be helpful to understand how Mathematica treats this integral.