# Inspired by Andrew Ellinor

Geometry Level 4

$\Large \mathfrak P=\displaystyle\int_0^{\pi/2}\mathfrak{T}(\phi)\mathrm{d}\phi$

Given $$\large \mathfrak{T}(\phi)=\displaystyle\sum_{n=0}^{\infty}\dfrac{\cos^3\phi}{\csc^{4n} \phi},~(|\csc \phi|\neq 1)$$, if the value of $$\mathfrak P$$ can be written in the form $$\large\dfrac{\pi}{\psi^{\psi}}$$ $$~~(\psi\in\mathbb{Z}^+)$$ , then: $\Large\color{red}{\psi^{\psi^{\psi+1}}}=\ ?$

Inspiration

×