Simplify:
$\;$ $\left(\dfrac{2}{x^2 - a^2}\right)^{-1} \left\{\left[\dfrac{x \left(x^2 - a^2\right)^{\frac{-1}{2}} + 1}{a \left(x - a\right)^{\frac{-1}{2}} + \left(x - a\right)^{\frac{1}{2}}} : \left(\dfrac{x - \left(x^2 - a^2\right)^{\frac{1}{2}}}{a^2 \left(x + a\right)^{\frac{1}{2}}}\right)^{-1} \right] + \left(x^2 + ax\right)^{-1} \right\}$
The given expression
$\left(\dfrac{2}{x^2 - a^2}\right)^{-1} \left\{\left[\dfrac{x \left(x^2 - a^2\right)^{\frac{-1}{2}} + 1}{a \left(x - a\right)^{\frac{-1}{2}} + \left(x - a\right)^{\frac{1}{2}}} : \left(\dfrac{x - \left(x^2 - a^2\right)^{\frac{1}{2}}}{a^2 \left(x + a\right)^{\frac{1}{2}}}\right)^{-1} \right] + \left(x^2 + ax\right)^{-1} \right\}$ $\;\;\; \cdots \; (1)$
Consider $\;\;\;$ $\dfrac{x \left(x^2 - a^2\right)^{\frac{-1}{2}} + 1}{a \left(x - a\right)^{\frac{-1}{2}} + \left(x - a\right)^{\frac{1}{2}}} : \left(\dfrac{x - \left(x^2 - a^2\right)^{\frac{1}{2}}}{a^2 \left(x + a\right)^{\frac{1}{2}}}\right)^{-1}$
$= \dfrac{\dfrac{x}{\sqrt{x^2 - a^2}} + 1}{\dfrac{a}{\sqrt{x - a}} + \sqrt{x - a}} : \dfrac{a^2 \sqrt{x + a}}{x - \sqrt{x^2 - a^2}}$
$= \dfrac{\left(x + \sqrt{x^2 - a^2}\right) \sqrt{x - a}}{\left(a + x - a\right) \sqrt{x^2 - a^2}} : \dfrac{a^2 \sqrt{x + a}}{x - \sqrt{x^2 - a^2}}$
$= \dfrac{\left(x + \sqrt{x^2 - a^2}\right) \sqrt{x - a}}{x \sqrt{x + a} \sqrt{x - a}} \times \dfrac{\left(x - \sqrt{x^2 - a^2}\right)}{a^2 \sqrt{x + a}}$
$= \dfrac{x^2 - x^2 + a^2}{a^2 x \left(x + a\right)}$
$= \dfrac{a^2}{a^2 x \left(x + a\right)}$
$= \dfrac{1}{x \left(x + a\right)}$ $\;\;\; \cdots \; (2)$
In view of expression $(2)$, the expression
$\left[\dfrac{x \left(x^2 - a^2\right)^{\frac{-1}{2}} + 1}{a \left(x - a\right)^{\frac{-1}{2}} + \left(x - a\right)^{\frac{1}{2}}} : \left(\dfrac{x - \left(x^2 - a^2\right)^{\frac{1}{2}}}{a^2 \left(x + a\right)^{\frac{1}{2}}}\right)^{-1} \right] + \left(x^2 + ax\right)^{-1}$
becomes
$= \dfrac{1}{x \left(x + a\right)} + \left(x^2 + ax\right)^{-1}$
$= \dfrac{1}{x \left(x + a\right)} + \dfrac{1}{x^2 + ax}$
$= \dfrac{2}{x \left(x + a\right)}$ $\;\;\; \cdots \; (3)$
$\therefore \;$ In view of expression $(3)$, expression $(1)$ becomes
$\left(\dfrac{2}{x^2 - a^2}\right)^{-1} \times \dfrac{2}{x \left(x + a\right)}$
$= \dfrac{\left(x + a\right) \left(x - a\right)}{2} \times \dfrac{2}{x \left(x + a\right)}$
$= \dfrac{x - a}{x}$