I found the expression of $\pi\varphi$ and $\pi/\varphi$ on Twitter via @AnecdotesMaths (https://twitter.com/AnecdotesMaths/status/1241032301783461890) in terms of an integral, and I wondered if anyone knew a proof for this ? Have a nice day !
$$\pi\times\varphi=\int_0^\infty\frac{5}{1+x^{10}} dx.$$ $$\frac{\pi}{\varphi}=\int_0^\infty\frac{5x^2}{1+x^{10}} dx.$$