Answer
$\frac{1}{\sqrt{x^{2}+1}+x}$
Work Step by Step
We simplify the fraction by multiplying through by $\sqrt{x^2+1}+x$ and using the fact that $(a-b)(a+b)=a^2-b^2$:
$\displaystyle \sqrt{x^{2}+1}-x=\frac{\sqrt{x^{2}+1}-x}{1} \displaystyle *\frac{\sqrt{x^{2}+1}+x}{\sqrt{x^{2}+1}+x}=\frac{x^{2}+1-x^{2}}{\sqrt{x^{2}+1}+x}=\frac{1}{\sqrt{x^{2}+1}+x}$
