Does anybody have an idea how to solve the integral
$$I(a,b)=\int_{-\pi}^{\pi} \mathrm{H}_n^{1} \left( \frac{a\,b}{\sqrt{\left(b\,\cos(\varphi)\right)^2+\left(a\,\sin(\varphi)\right)^2}}\right) \, \cos(n\,\varphi) \, \mathrm{d}\varphi$$
with $a>0$, $b>0$ and $n\in \mathbb{N}_0$.
Also the the solution for $b\to 0$ would be interesting.
Thank you for any help.
A possible direction (too long for a comment):
Use the multiple angle formulas to express $\cos n\phi$ as a sum of powers, i.e.: $$\cos n\phi = \sum \alpha_k\cos^k(\phi)$$
Now let $x=\cos \phi$, so that $dx = -\sin \phi d\phi$. $$I(a,b)= \sum \alpha_k \int_{-\pi}^\pi H^1_n\left(\frac{ab}{(b^2 - a^2)x^2 + a^2}\right)\frac{x^k}{\sqrt{1 - x^2}}\ dx$$
Now, this still isn't very helpful, but perhaps you can use the integral form of $H_n^1$ and switch the integration order.