Integral $\int(\frac{1}{n+\cos x})^{\frac{3}{2}}dx$

Question

Find an expression for this indefinite integral.
I tried using some online calculators, but it is not coming.

$$\int\left(\dfrac{1}{n+\cos x}\right)^{\frac{3}{2}}dx$$

Answer 1

Hint:

• For $n=\pm1$, use the fact that $~\dfrac{1+\cos2t}2=\cos^2t~$ and $~\dfrac{1-\cos2t}2=\sin^2t$.

• For $n\neq\pm1$, use elliptic integrals. You should ultimately arrive at the following expression:

$\qquad\qquad$