Solving:
$$ \int \sqrt{4\sin^2(x) + \cos^2(x) } dx $$
I have used calculators attempt this problem, but they all use the elliptical integral of the second parameter. My goal is to somehow solve this without using it.
Trying some substitutions:
$u = \sin^2(x)$, $\frac{du}{2\cos(x)\sin(x)} = dx$
$\sin(x)$ = $\sqrt{u}$, $\cos(x)$ = $\sqrt{1-u}$,
$$ \int \frac{\sqrt{3u + 1 }}{2(\sqrt{1-u})\sqrt{u}} du = $$
$$ \int \frac{\sqrt{3u + 1 }}{2\sqrt{u-u^2 } } du = $$
$$ \frac{1}{2} \int \frac{\sqrt{3u + 1 }}{\sqrt{u-u^2 } } du $$
Any ideas on how to proceed?