\(\dfrac{x^3-x}{x^2+1}\cdot\left(\dfrac{1}{x^2-2x+1}+\dfrac{1}{1-x^2}\right)-\dfrac{x^2+x+1}{x^3-1}\\ =\dfrac{x\cdot\left(x^2-1\right)}{x^2+1}\cdot\left[\dfrac{1}{\left(x-1\right)^2}-\dfrac{1}{\left(x-1\right)\left(x+1\right)}\right]-\dfrac{x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}\\ =\dfrac{x\left(x-1\right)\left(x+1\right)}{x^2+1}\cdot\dfrac{\left(x+1\right)-\left(x-1\right)}{\left(x-1\right)^2\left(x+1\right)}-\dfrac{1}{x-1}\\ =\dfrac{x\left(x-1\right)\left(x+1\right)}{x^2+1}\cdot\dfrac{2}{\left(x-1\right)^2\left(x+1\right)}-\dfrac{1}{x-1}\\ =\dfrac{2x}{\left(x^2+1\right)\left(x-1\right)}-\dfrac{1}{x-1}\\ =\dfrac{2x-\left(x^2+1\right)}{\left(x^2+1\right)\left(x-1\right)}=\dfrac{-x^2+2x-1}{\left(x^2+1\right)\left(x-1\right)}\\ =-\dfrac{x^2-2x+1}{\left(x^2+1\right)\left(x-1\right)}=\dfrac{-\left(x-1\right)^2}{\left(x^2+1\right)\left(x-1\right)}\\ =-\dfrac{x-1}{x^2+1}=\dfrac{1-x}{x^2+1}\)
