Integration through partial fractions with complex roots In integrating the following: $\frac{1}{(x^2+2x+3)^2}$ I am trying to use partial fraction decomposition as follows:
$\frac{1}{(x^2+2x+3)^2} = \frac{Ax + B}{x^2+2x+3} + \frac{Cx + D}{(x^2+2x+3)^2}$
Which gives me:
$1 = (Ax + B)(x^2 +2x +3) + (Bx + C)$
And that gets me back to where I started.
$\frac{1}{(x^2+2x+3)^2}$
What am I doing wrong?

That is already written as partial fractions.
It seems you want:
$\begin{align*}
  \int \frac{d x}{(x^2 + 2 x + 3)^2}
    &= \frac{1}{\sqrt{2}} \int \frac{d u}{(u^2 + 1)^2} \qquad u = \sqrt{2} (x + 1)
\end{align*}$
This last one yields to a trigonometric substitution.
