Evaluate: ∫ (x²+x+1)/(x+2)(x²+1) dx (Evaluate)

Let $\frac{x^2+x+1}{(x+2)(x^2+1)} = \frac{A}{x+2} + \frac{Bx+C}{x^2+1}$. Equating coefficients: $A + B = 1$, $2B + C = 1$, $A + 2C = 1$, giving $A = \frac{3}{5}$, $B = \frac{2}{5}$, $C = \frac{1}{5}$. Hence $\int \left[\frac{3}{5(x+2)} + \frac{2x+1}{5(x^2+1)}\right]dx = \frac{3}{5}\ln|x+2| + \frac{1}{5}\ln(x^2+1) + \frac{1}{5}\tan^{-1}x + C$.