Non-Elementary Integration · Superchallenge problem

Given that $\displaystyle\int_0^{\infty}\dfrac{x^{s-1}}{1+x}\,dx=\dfrac{\pi}{\sin(\pi s)}$ for $0<s<1$, evaluate $\displaystyle\int_0^{\infty}\dfrac{dx}{x^{2/3}(1+x)}$.