Question Number 98776 by M±th+et+s last updated on 16/Jun/20
$$\int_{\mathrm{0}} ^{\infty} \frac{\left({x}−\mathrm{1}\right)}{{ln}\left({F}\left({x}\right)\sqrt{\mathrm{5}}+{cos}\left(\pi{x}\right)\left(\varphi\right)^{−{x}} −\mathrm{1}\right)\sqrt{{F}\left({x}\right)\sqrt{\mathrm{5}}+{cos}\left(\pi{x}\right)\left(\varphi\right)^{−{x}} −\mathrm{1}}}{dx} \\ $$$$ \\ $$$${F}\left({x}\right)={Fib}\left({x}\right)={xth}\:{Extended}\:{fibonacci}\:{number} \\ $$$${f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$$\varphi=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$