Evaluate: ∫ (dx)/(x³ + x² + x + 1)

$x^3 + x^2 + x + 1 = x^2(x+1) + (x+1) = (x+1)(x^2+1)$ Let $\frac{1}{(x+1)(x^2+1)} = \frac{A}{x+1} + \frac{Bx + C}{x^2+1}$, so $1 = A(x^2+1) + (Bx+C)(x+1)$. $x = -1$: $A = \frac{1}{2}$. Coefficient of $x^2$: $A + B = 0$, so $B = -\frac{1}{2}$. Constant: $A + C = 1$, so $C = \frac{1}{2}$. $\int \frac{dx}{x^3+x^2+x+1} = \frac{1}{2}\int \frac{dx}{x+1} - \frac{1}{2}\int \frac{x\,dx}{x^2+1} + \frac{1}{2}\int \frac{dx}{x^2+1}$ Hence $= \frac{1}{2}\log|x+1| - \frac{1}{4}\log(x^2+1) + \frac{1}{2}\tan^{-1} x + C$.