\(\left[\left(\dfrac{1}{a^2+1}\right).\dfrac{1}{a^2+2a+1}+\dfrac{2}{\left(a+1\right)^3}.\left(\dfrac{1}{a}+1\right)\right]:\dfrac{a-1}{a^3}\)
\(=\left(\dfrac{1}{\left(a^2+1\right)\left(a^2+2a+1\right)}+\dfrac{2}{\left(a+1\right)^3}\cdot\dfrac{1+a}{a}\right)\cdot\dfrac{a^3}{a-1}\)
\(=\left(\dfrac{1}{\left(a^2+1\right)\left(a+1\right)^2}+\dfrac{2}{a\left(a+1\right)^2}\right)\cdot\dfrac{a^3}{a-1}\)
\(=\dfrac{a+2\left(a^2+1\right)}{a\left(a^2+1\right)\left(a+1\right)^2}\cdot\dfrac{a^3}{a-1}\)
\(=\dfrac{a+2a^2+2}{\left(a^2+1\right)\left(a+1\right)^2}\cdot\dfrac{a^2}{a-1}\)
\(=\dfrac{a^2\left(a+2a^2+2\right)}{\left(a^2+1\right)\left(a+1\right)^2\cdot\left(a-1\right)}\)
\(=\dfrac{a^3+2a^4+2a^2}{\left(a^3-a^2+a-1\right)\left(a+1\right)^2}\)