I need help with the calculation of this integral: $$\vec H(r,a) = \int_{0}^{2\pi}\sin(\psi)\frac{\rho e^{i\psi}-re^{ia}}{(\rho^2+r^2-2\rho r\cos(\psi - a))^{3/2}} d\psi$$
Equivalently I want to know these 2 integrals: $$ f(r,a)=\int_{0}^{2\pi}\sin(\psi)\frac{\rho \cos(\psi)-r\cos(a)}{(\rho^2+r^2-2\rho r\cos(\psi - a))^{3/2}}d\psi$$
and $$ g(r,a)=\int_{0}^{2\pi}\sin(\psi)\frac{\rho \sin(\psi)-r \sin(a)}{(\rho^2+r^2-2\rho r\cos(\psi - a))^{3/2}}d\psi$$ i.e. $\vec H(r,a) = f(r,a) + i \cdot g(r,a)$
I assume that $\rho = ct$ and $\rho > r > 0$. Also $a \in [0,2\pi)$
Wolfram Alpha could not calculate it. If anyone knows a methodology for this or the answer I would be grateful.
However, if anyone can prove that for $r = ct$ $$\vec H(r,a) = \vec H(r,a + k*pi) \space, \forall k \in \mathbb{Z}$$ I would be partially ok.
Thank you!