Is there a sensible analytic solution to the following integral: $$\int_a^b \frac{\arctan(A+Bt)}{C^2 + (t-Z)^2}dt$$ where all constants are real and $C>0$.
This integral is part of the third term in a Neumann expansion applied on a specific Rendering equation. I have solved the first and second terms in 2D and 3D, an interactive implementation can be found here (shadertoy).
Mathematica was used first and it outputs an expression for the integral with sums of complex logs and polylogs (I use assumptions to help the program). The log-sum is multiplied by $i$, so all real parts of the log-sum should cancel out since the integrand is real. As far as I know there is no way to generally extract the imaginary part from $\operatorname{Li}_2(\ldots)$. No practical implementation probably exists based on this expression.
I have also tried to use Feynman's integral trick, but not having much success. Since Mathematica is limited (the following example fails: $\int_0^\pi\ln(1-2a\cos(x)+a^2)\,dx$ from this tutorial, which has a closed form), I hoper there is a simpler solution without the dilogarithm.
Edit: clarified the integral