I am considering the following integral $$\int_{-t}^{\infty} \frac{dy}{y-s} \frac{1}{y^2} \frac{1}{y^{\epsilon}} {}_2F_1(1,1,2+\epsilon, -t/y)$$ Rewriting the hypergeometric using its integral representation and making a change of variables $y=-t/u$ I obtain the integral, up to some numerical factors, $$\int_0^1 \int_0^1 dz\, du (1-uz)^{-1}u^{1+\epsilon} (1-z)^{\epsilon} (1+\frac{us}{t})^{-1}$$ My question is how to make progress with this? I have attempted partial fractions on the term $$\frac{1}{(1-uz)} \frac{1}{(1+\frac{us}{t})} = \frac{z}{\frac{s}{t}+z} \frac{1}{(1-uz)} + \frac{\frac{s}{t}}{\frac{s}{t}+z}\frac{1}{(1+\frac{us}{t})}$$ but I am not sure if this has helped me at all.
Thanks for any comments!