Split a numerator into a denominator derivative and a remainder

Compare coefficients: $A=1/2$, and $2A+B=3$ gives $B=2$.

Split the numerator to get $\frac12\int\dfrac{2x+2}{x^2+2x+2}dx+2\int\dfrac1{1+(x+1)^2}dx$. This is $\frac12\ln|x^2+2x+2|+2\tan^{-1}(x+1)+C$. The quadratic is always positive, so the bars are optional.