The function I'm looking to find is: $$ \sigma = \int_{-a}^{a} \frac{(x-x')((x-x')^2 - y^2)}{((x-x')^2 + y^2)^2} \ln \left|\frac{x(a^2 - c^2)^{1/2} + c(a^2 - x^2)^{1/2}}{x(a^2 - c^2)^{1/2} - c(a^2 - x^2)^{1/2}}\right| dx$$
Analytically there aren't any obvious ways to solve this integral, so I attempted to do it using mathematica but it simply gave up - I tried to separate it into two integrals by splitting up the log expression into two terms but still did not succeed.
I am now looking at possible numerical methods to try and extract some kind of closed form of this expression but I am very new to numerical integration and was under the impression this would not be possible. I'm absolutely stuck on how to evaluate this integral, let alone if it is possible (evaluating this integral expression is extremely central to the rest of my project, unfortunately!); any pointers would be very much appreciated.
