Simplify: $\;$ $\left(\dfrac{a + b}{a - b}\right)^{\frac{1}{2}} - \dfrac{2a \sqrt{a^2 - b^2}}{b^2 \left(ab^{-1} + 1\right)^2} \times\dfrac{1}{1 + \dfrac{1 - b a^{-1}}{1 + ba^{-1}}}$
$\left(\dfrac{a + b}{a - b}\right)^{\frac{1}{2}} - \dfrac{2a \sqrt{a^2 - b^2}}{b^2 \left(ab^{-1} + 1\right)^2} \times\dfrac{1}{1 + \dfrac{1 - b a^{-1}}{1 + ba^{-1}}}$
$= \dfrac{\left(\sqrt{a + b}\right) \left(\sqrt{a + b}\right)}{\left(\sqrt{a - b}\right) \left(\sqrt{a + b}\right)} - \dfrac{2a \sqrt{a^2 - b^2}}{b^2 \left(\dfrac{a}{b} + 1\right)^2} \times \dfrac{1}{1 + \dfrac{1 - \dfrac{b}{a}}{1 + \dfrac{b}{a}}}$
$= \dfrac{a + b}{\sqrt{a^2 - b^2}} - \dfrac{2a \sqrt{a^2 - b^2}}{\left(a + b\right)^2} \times \dfrac{1}{1 + \dfrac{a - b}{a + b}}$
$= \dfrac{a + b}{\sqrt{a^2 - b^2}} - \dfrac{2a \sqrt{a^2 - b^2}}{\left(a + b\right)^2} \times \dfrac{a + b}{a + b + a - b}$
$= \dfrac{a + b}{\sqrt{a^2 - b^2}} - \dfrac{2a \sqrt{a^2 - b^2}}{a + b} \times \dfrac{1}{2a}$
$= \dfrac{a + b}{\sqrt{a^2 - b^2}} - \dfrac{\sqrt{a^2 - b^2}}{a + b}$
$= \dfrac{\left(a + b\right)^2 - \left(a^2 - b^2\right)}{\left(a + b\right) \sqrt{a^2 - b^2}}$
$= \dfrac{a^2 + b^2 + 2ab - a^2 + b^2}{\left(a + b\right) \sqrt{a^2 - b^2}}$
$= \dfrac{2b^2 + 2ab}{\left(a + b\right) \sqrt{a^2 - b^2}}$
$= \dfrac{2b \left(a + b\right)}{\left(a + b\right) \sqrt{a^2 - b^2}}$
$= \dfrac{2b}{\sqrt{a^2 - b^2}}$