$\frac{\sqrt{a^2+1}-a}{bc}=\frac{\sqrt{(a+b)(a+c)}-a}{bc}\leq \frac{b+c}{2bc}=\frac{1}{2}(\frac{1}{b}+\frac{1}{c})$( AM-GM)$\Rightarrow \sum_{cyc}^{}\frac{\sqrt{a^2+1}-a}{bc}\leq \frac{1}{a}+\frac{1}{b}+\frac{1}{c} $
"=" khi $a=b=c=\frac{\sqrt{3}}{3}$