smc

We rationalize the denominator in the given equation, then solve for $x:$ $$\begin{array}{rl}x+\sqrt{x^2-1}+\frac{x+\sqrt{x^2-1}}{\left(x-\sqrt{x^2-1}\right)\left(x+\sqrt{x^2-1}\right)}& =20\\ x+\sqrt{x^2-1}+x+\sqrt{x^2-1}& =20\\ x+\sqrt{x^2-1}& =10\\ \sqrt{x^2-1}& =10-x\\ x^2-1& =100-20x+x^2\\ 20x& =101\\ x& =\mathrm{5.05.}\end{array}$$ We rationalize the denominator in the requested expression, then simplify the result: $$\begin{array}{rl}{x}^2+\sqrt{x^4-1}+\frac{1}{x^2+\sqrt{x^4-1}}& =x^2+\sqrt{x^4-1}+\frac{x^2-\sqrt{x^4-1}}{\left(x^2+\sqrt{x^4-1}\right)\left(x^2-\sqrt{x^4-1}\right)}\\ & =x^2+\sqrt{x^4-1}+x^2-\sqrt{x^4-1}\\ & =2x^2\\ & =\boxed{51.005}.\end{array}$$