Bern's Integral

$\begin{aligned} A & = \int_1^2 \ln^2 \left(x+\sqrt{x^2-1}\right) dx & \small \color{#3D99F6} \text{Let }u = \sqrt{x^2-1} \implies du = \frac x{\sqrt{x^2-1}} dx \\ & = \int_0^{\sqrt 3} \frac {u\ln^2 \left(\sqrt{u^2+1} +u\right)}{\sqrt{u^2+1}} du & \small \color{#3D99F6} \text{By integration by parts.} \\ & = \sqrt{u^2+1}\ln^2 \left(\sqrt{u^2+1} +u\right) \bigg|_0^{\sqrt 3} - 2 \int_0^{\sqrt 3} \ln \left(\sqrt{u^2+1} +u\right) du & \small \color{#3D99F6} \text{Repeat integration by parts.} \\ & = 2 \ln^2 \left(2+\sqrt 3\right) - 2u \ln \left(\sqrt{u^2+1} +u\right) \bigg|_0^{\sqrt 3} + 2 \int_0^{\sqrt 3} \frac u{\sqrt{u^2+1}} du \\ & = 2 \ln^2 \left(2+\sqrt 3\right) - 2\sqrt 3 \ln \left(2+\sqrt 3\right) + 2 \sqrt{u^2+1} \bigg|_0^{\sqrt 3} \\ & = 2 \ln^2 \left(2+\sqrt 3\right) - 2\sqrt 3 \ln \left(2+\sqrt 3\right) + 2 (2-1) \\ & \approx 0.906680227 \end{aligned}$

$\implies \lfloor 1000A \rfloor = \boxed{906}$