\begiin{align}
&\Rightarrow \sqrt{\frac{1-a_{1}}{a_{3}-1}}+\sqrt{\frac{a_{3}-1}{1-a_{1}}}=\sqrt{\frac{\sqrt{2}(\sqrt{2}+1) \ln (\sqrt{2}+1)}{\sqrt{2}(\sqrt{2}-1) \ln (\sqrt{2}+1)}}+\sqrt{\frac{\sqrt{2}(\sqrt{2}-1) \ln (\sqrt{2}+1)}{\sqrt{2}(\sqrt{2}+1) \ln (\sqrt{2}+1)}}\\
&=\sqrt{\frac{\sqrt{2}+1}{\sqrt{2}-1}}+\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}=\sqrt{2}+1+\sqrt{2}-1=2 \sqrt{2}
\end{align}