How to solve this integral

Question

Wolfram integrator is unable to solve this:

$$ \int_0^\pi \frac{1}{\sqrt {(a-\cos(t))^2+(b-\sin(t))^2+c^2}}\,\mathrm dt $$

Any suggestions? Thanks!

Answer 1

Note that $(a-\cos(t))^2 + (b-\sin(t))^2 + c^2 = a^2 + b^2 + c^2 + 1 - 2(a \cos(t) + b \sin(t))$.

Also note that $a \cos (t) + b \sin(t) = \sqrt{a^2 + b^2} \sin (t + F)$.

Let $d = a^2 + b^2 + c^2 + 1$ and $e = 2\sqrt{a^2 + b^2}$. Then, our integral becomes

$\displaystyle\int_0^\pi \frac{1}{\sqrt{d - e \sin(t + F)}}\mathrm dt$, where $d,e$, and $F$ are constants. This is elliptic and has no simple solution in terms of elementary functions.

Answer 2

This can be rewritten

$ \int_0^{\pi} \frac{1}{\sqrt{a^2 + b^2 + c^2 +1 - 2a \cos t - 2b \sin t}} dt $

which has the form

$ A \int_0^{\pi} \frac{1}{\sqrt{1 + B \sin(t+\phi)}} dt $

for appropriate constants $A$, $B$, and $\phi$. I just tried the indefinite integral

$ \int \frac{1}{\sqrt{1 + B \sin(t+\phi)}} dt $

in Wolfram Alpha. It provided a closed form answer.

I hope this helps.