In comments to this question, @RobertIsrael asserted that, for $-1, $ \int_0^{2\pi} \frac{1-x \cos(\phi)}{\left(1 - 2 x \cos(\phi) + x^2\right)^{3/2}} \mathrm{d} \phi = \frac{4}{1-x^2} \operatorname{E}(x^2) \tag{1} $ where $E(m)$ is the complete elliptic integral of the second kind: $E(m) = \int_0^{\pi/2} \sqrt{1-m \sin^2(\theta)} \mathrm{d} \theta$.
It is easy to verify that the series expansion of the integrand, integrated term-wise, agrees with the series expansion of Robert's elegant answer.
I am very much interested if there is a way to directly establish $\text{eq. (1)}$ from the integral.
special-functions
definite-integrals