It's me again.
This time I would like to ask you whether you know if there is some closed expression for the following integral: $$\int \frac{-s}{1+\epsilon s+s^2} ds$$ So far I thought about how we can rewrite it as $$-\int \frac{y-\frac{\epsilon}{2}}{[1-(\frac{\epsilon}{2})^2] + y^2} dy$$
if we set $y=s+\frac{\epsilon}{2}$. Maybe some convenient functions can do the trick, but I am not sure what to try next.
Thanks in advance!
You need to use this substitution first: $x=y^2+\left[1-\frac{\epsilon^2}4\right]$. Then $dx=2ydy$. This yields one integral of type $$\int\frac{dx}x$$and one integral of type $$\int\frac{dx}{x^2+a^2}$$ These are standard integrals.