ELLIPTIC INTEGRAL HYPERBOLIC FUNCTIONS

Today we will prove the following result that appears in this Twitter post




\begin{align*} \int_0^{\pi/2}\frac{d \theta}{\left(\frac13 + \sin^2(\theta) \right)^{1/3}}&=\frac{\sqrt[3]{3}}{\sqrt[3]{2^7}}\cdot\frac{\Gamma^3\left(\frac13\right)}{\pi} \end{align*}



Click here to see the proof.

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