TITLE: How to compute $\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$?
QUESTION [29 upvotes]: How to compute the integral,
$$\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$$
where, $\varphi = \dfrac{\sqrt{5}+1}{2}$ is the Golden Ratio?

REPLY [2 votes]: For an answer avoiding the use of special functions: use $w=[x^{\varphi}(1+x^{\varphi})^{-1}]^{\varphi}$ then $$\int_0^1 \mathrm{d}w = 1$$