Integrating $\int_{-\pi}^{\pi}\frac{-8 (\theta - cos\lambda)^3}{(1-2\theta cos\lambda +\theta^2)^3}d\lambda$

Question

Can anyone help me with solving the following intergral.

\begin{align*} \int_{-\pi}^{\pi}\frac{-8 (\theta - cos\lambda)^3}{(1-2\theta cos\lambda +\theta^2)^3}d\lambda \end{align*}

Any hints also appreciated.

Answer 1

$\newcommand{\mmphyres}{\displaystyle\operatorname*{Res}}$ Substitution $z=e^{i\lambda}$ and the residue theorem give $$\int_{-\pi}^{\pi}\left(\frac{2\theta-2\cos\lambda}{1-2\theta\cos\lambda+\theta^2}\right)^3\,d\lambda=2\pi\begin{cases}\mmphyres_{z=0}f+\mmphyres_{z=\theta}f, & |\theta|<1 \\ \mmphyres_{z=0}f+\mmphyres_{z=1/\theta}f, & |\theta|>1\end{cases}$$ where $f(z)=\displaystyle\frac{1}{z}\left(\frac{1-2\theta z+z^2}{(\theta-z)(1-\theta z)}\right)^3$. Fairly boring calculations show that $$\mmphyres_{z=0}f=\dfrac{1}{\theta^3},\quad\mmphyres_{z=1/\theta}f=-\mmphyres_{z=\theta}f=\frac{1}{\theta^3}+\frac{6\theta}{(1-\theta^2)^2}.$$