I need to find a primitive of
$$\sqrt{4 - \cos^2\!x \over 1 + 4\,\cos x}$$
or alternatively of
$$\sqrt{4 - v^2 \over (1 + 4 v)(1 - v^2)}.$$
Not in terms of elementary functions. Even special functions would be fine. Or at least the definite integral for $x$ from 0 to $\arccos(-1/4)$ (for $v$ from -1/4 to 1).
Thanks.