Find $\int \sin(2x)\sqrt{1+3\cos^{2}(x)}\ dx$ Is it humanly possible to integrate the following equation?
$$\int \sin(2x)\sqrt{1+3\cos^{2}(x)}\ dx$$

Let $u=\cos^2(x)$, therefore $du=-2\cos(x)\sin(x)\ dx$:
$$\int\sin(2x)\sqrt{1+3\cos^2(x)}\ dx=-\int\sqrt{1+3u}\ du$$
Now, let $v=1+3u$ and $dv=3\ du$:
\begin{align*}
-\int\sqrt{1+3u}\ du&=-\frac13\int\sqrt v\ dv\\
&=-\frac{2v^{3/2}}9+C,\text{ where $C$ is a constant}\\
&=-\frac29(1+3u)^{\frac32}+C\\
&=\boxed{-\frac29(1+3\cos^2(x))^{\frac32}+C}
\end{align*}
