This should've been mechanics!

Calculus Level 5

Consider a mathematical pendulum. It is held at a position as shown above at \(t=0\), before it is released. Find the time taken by it to complete one oscillation, that is the time taken by it to make a complete round from its initial position to back to the initial position. If the time taken can be represented as:

\[ \sqrt{\dfrac{L^{\alpha}}{\pi^{\beta}g^{\gamma}}}\Gamma^{\delta}\left(\dfrac{1}{\eta}\right) \]

where \(\Gamma\) is the gamma function and \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), and \(\eta \) are all positive integers. Find \(\alpha +\beta +\gamma +\eta^{\delta}\).

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