少しだけ難しい有理化問題

\begin{align*} \frac{1}{2\sqrt{2}+\sqrt{3}+1} & =\frac{\left(2\sqrt{2}-\left(\sqrt{3}+1\right)\right)}{\left(2\sqrt{2}+\sqrt{3}+1\right)\left(2\sqrt{2}-\left(\sqrt{3}+1\right)\right)}\\ & =\frac{2\sqrt{2}-\left(\sqrt{3}+1\right)}{\left(2\sqrt{2}\right)^{2}-\left(\sqrt{3}+1\right)^{2}}\\ & =\frac{2\sqrt{2}-\sqrt{3}-1}{8-\left(3+1+2\sqrt{3}\right)}\\ & =\frac{2\sqrt{2}-\sqrt{3}-1}{4-2\sqrt{3}}\\ & =\frac{\left(2\sqrt{2}-\sqrt{3}-1\right)\left(4+2\sqrt{3}\right)}{\left(4-2\sqrt{3}\right)\left(4+2\sqrt{3}\right)}\\ & =\frac{8\sqrt{2}-6\sqrt{3}+4\sqrt{6}-10}{4^{2}-\left(2\sqrt{3}\right)^{2}}\\ & =\frac{8\sqrt{2}-6\sqrt{3}+4\sqrt{6}-10}{4}\\ & =\frac{4\sqrt{2}-3\sqrt{3}+2\sqrt{6}-5}{2} \end{align*}