Assistance with solving the integral Can you give me an idea how to handle this integral?
$\int_{0}^{2\pi} (1+\cos{x})\sqrt{3+\cos{x}}\,dx$

Using the tangent half-angle substitution
$$I=\int_{0}^{2\pi} (1+\cos(x))\sqrt{3+\cos(x)}\,dx=2\int_{0}^{\pi} (1+\cos(x))\sqrt{3+\cos(x)}\,dx$$
$$I=8 \sqrt{2} \int_0^\infty \sqrt{\frac{t^2+2}{\left(t^2+1\right)^5}}\,dt=4 \sqrt{2}\int_1^\infty \frac 1 {x^2}\sqrt{\frac{x+1}{(x-1) x}}$$ The antiderivative is  nasty but its evaluation at $\infty$ is simply
$$I=\frac{2 }{3 \sqrt{\pi }}\left(\Gamma \left(\frac{1}{4}\right)^2+12\, \Gamma
   \left(\frac{3}{4}\right)^2\right)$$ which is the same as
$$I=16  \left(E\left(\frac{1}{2}\right)-\frac{1}{3}K\left(\frac{1}{2}\right)\right)$$ and
$$I=8 \sqrt{2} \left(E(-1)-\frac{2 }{3}K(-1)\right)$$ already given by @Robert Lee.
