Integrating Annoying Trig Integral: Help Appreciated

[radiogaga35]

Hi...would appreciate any suggestions re the following integral which has appeared in a celestial-mechanics calculation:

$$I = \int_0^{2\pi } {\frac{1}{{(1 + e\cos \theta )^3}}d\theta }$$

where $$0 < e < 1$$.

Integration by parts seems a sensible approach but for some reason I can't get sensible results. I presume I'm making some idiotic mistake that I'm just not picking up when I check my calculations (frustrating as hell!). I'm pretty sure there is supposed to be a fairly neat result but MATLAB and Mathematica aren't giving me anything.

Any ideas? Thanks in advance.

 

[Count Iblis]

Apply the conformal transform z = exp(i theta). Then the integral becomes the integral over the unit circle in the complex plane of:


dz/(i z) 1/[1+e (z+z^(-1))/2]^3

Simplify the fraction and consider the poles that are inside the unit circle.

 

[radiogaga35]

Thank you!