@Axel Vogt Thank you!

I changed the range and saw there is a huge region where it is imaginary. This does confuse me about the physics going on here as it is supposed to represent a real-measurable value. If anyone is interested in the paper I got this equation from its "On the Probability of Magnification by Cosmologically Distributed Gravitational Lenses".

Thank you all again for your help!

@Carl Love 

Hi, I'm unsure why you are only considering a single component of the whole integral, could you walk me through that? 

Also, I plotted the function that is being integrated and the graph is positive and real for all values of "r" so it's integral should also be real for real values of "r" right?: 

Here is the Maple code:

restart;
with(plots);
Digits := 30;
E := 3.83*10^14;
R := 0.1*E;
l := 0.01*R;
f := (-4*E^2 + r^2)^(1/2)*(R^2 - l^2 + r^2)/((R^2 - (l - r)^2)^(1/2)*((l + r)^2 - R^2)^(1/2));
plot(f, r = -R .. R, labels = ["r", " f(r)"], labeldirections = ["horizontal", "vertical"]);