Answer
$A=a, B=b, C=-a, D=-b$
Work Step by Step
Step 1. Combine the right side rational functions:
RHS=$\frac{(Ax+B)(x^2+1)+(Cx+D)}{(x^2+1)^2}=\frac{(A)x^3+(B)x^2+(A+C)x+(B+D)}{(x^2+1)^2}$
Step 2. Compare the above results with the expression on the left side, we get:
\begin{cases} A=a \\ B=b\\A+C=0\\B+D=0 \end{cases}
Step 3. Use substitution to get: $A=a, B=b, C=-a, D=-b$
