$${\left(\frac{a^{-1}+b^{-1}}{b{a}^{-1}+a{b}^{-1}}\right)}^{-1}+{\left(\frac{a^{-1}+b^{-1}}{2}\right)}^{-1}-\frac{a^{-1}-b^{-1}}{a^{-1}·b^{-1}}=$$

$$=\frac{b{a}^{-1}+a{b}^{-1}}{a^{-1}+b^{-1}}+\frac{2}{a^{-1}+b^{-1}}-\frac{{\displaystyle \frac{1}{a}}-{\displaystyle \frac{1}{b}}}{{\displaystyle \frac{1}{ab}}}$$

$$=\frac{{\displaystyle \frac{b}{a}}+{\displaystyle \frac{a}{b}}}{{\displaystyle \frac{1}{a}}+{\displaystyle \frac{1}{b}}}+\frac{2}{{\displaystyle \frac{1}{a}}+{\displaystyle \frac{1}{b}}}-\frac{{\displaystyle \frac{b-a}{ab}}}{{\displaystyle \frac{1}{ab}}}$$

$$=\frac{{\displaystyle \frac{b^2+a^2}{ab}}}{{\displaystyle \frac{b+a}{ab}}}+\frac{2}{{\displaystyle \frac{b+a}{ab}}}-\frac{\cancel{ab}\left(b-a\right)}{\cancel{ab}}$$

$$=\frac{ab\left(a^2+b^2\right)}{ab\left(a+b\right)}+\frac{2ab}{a+b}+a-b$$

$$=\frac{ab\left(a^2+b^2\right)+2{\left(ab\right)}^2+ab\left(a+b\right)\left(a-b\right)}{ab\left(a+b\right)}$$

$$=\frac{ab\left[\left(a^2+b^2\right)+2ab+\left(a+b\right)\left(a-b\right)\right]}{ab\left(a+b\right)}$$

$$=\frac{\cancel{ab}\left[\left(a^2+b^2\right)+2ab+\left(a+b\right)\left(a-b\right)\right]}{\cancel{ab}\left(a+b\right)}$$

$$=\frac{{\left(a+b\right)}^2+\left(a+b\right)\left(a-b\right)}{a+b}$$

$$=\frac{\left(a+b\right)\left(a+b+a-b\right)}{\left(a+b\right)}$$

$$=\frac{\cancel{\left(a+b\right)}\left(a\cancel{+b}+a\cancel{-b}\right)}{\cancel{\left(a+b\right)}}$$