Integration by using special functions $$\int ^{\pi }_{0}\dfrac {dt}{\sqrt {3-\cos t}}$$
How can you solve the following equation by using alpha/gamma functions and putting
$$\cos t=1-2\sqrt {u}$$
 A: $\newcommand{\+}{^{\dagger}}
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\begin{align}
\color{#00f}{\large\int_{0}^{\pi}{\dd t \over \root{3 - \cos\pars{t}}}}&=
\int_{0}^{\pi}{\dd t \over \root{3 - \bracks{1 - 2\sin^{2}\pars{t/2}}}}
=\int_{0}^{\pi}{\dd t \over \root{2 + 2\sin^{2}\pars{t/2}}}
\\[3mm]&=2\int_{0}^{\pi/2}{\dd t \over \root{2 + 2\sin^{2}\pars{t}}}
=\root{2}\int_{0}^{\pi/2}{\dd t \over \root{1 + \sin^{2}\pars{t}}}
\\[3mm]&=\color{#00f}{\large\root{2}\,{\rm K}\pars{-1}}
\end{align}
where
$\ds{{\rm K}\pars{m} = \int_{0}^{\pi/2}{\dd t \over \root{1 - m\,\sin^{2}\pars{t}}}}$
is the Elliptic Integral of the First Kind.
A: Maple does this using EllipticK:
> int(1/sqrt(3-cos(t)),t=0..Pi);
$$  \text{EllipticK}\left(\dfrac{1}{2}\sqrt{2}\right)$$
> select(has,[FunctionAdvisor(specialize,%)],GAMMA);
$$
[[{\rm EllipticK} \left( \dfrac12\sqrt {2} \right) =\dfrac12{\frac {{\pi }^{
3/2}}{  \Gamma  \left( 3/4 \right)   ^{2}}},\mbox { 
`with no restrictions`}]]
$$
