In this answer to How is the average distance between 2 objects orbiting around a third object calculated? I had to integrate
$$\int_0^{2 \pi}\sqrt{(a-\cos \theta)^2 + \sin^2 \theta} \ d\theta.$$
I tried to find the integral analytically with Wolfram Alpha but it returned an error message:
Standard computation time exceeded…
which surprised me; I'd figured that this was known and easily looked-up by the site.
Does this mean that there is no known analytical form for this definite integral? Or for some reason is it particularly challenging?
Best Answer
It is not an error message but just "Standard computation time exceeded".
What you should have obtained is $$I=\int_0^{2 \pi}\sqrt{(a-\cos (\theta))^2 + \sin^2 (\theta)} \ d\theta=$$ $$I=2 \left(\sqrt{(a-1)^2} E\left(-\frac{4 a}{(a-1)^2}\right)+\sqrt{(a+1)^2} E\left(\frac{4 a}{(a+1)^2}\right)\right)$$ provided, if $a$ is a real, that $$\Re(a (a+2))>-1\land \Re((a-2) a)>-1$$ where appear ellptic integrals of the second kind. In fact, this reduces to $$I=4(a+1)E\left(\frac{4 a}{(a+1)^2}\right)$$