Rational Function Integration

Question

This looks to be a simple problem, but it has me stumped. I already have the answer, but a step-by-step solution would be appreciated.

$$\int\frac{x+4}{x^2+2x+5}$$

Answer 1

To elaborate on my comments: $$\begin{align*} \int\frac{x+4}{(x+1)^2+4}\,dx&=\int\frac{x+1}{(x+1)^2+4}\,dx+\int\frac{3}{(x+1)^2+4}\,dx\\[1ex] &=\frac{1}{2}\int\frac{2x+2}{(x+1)^2+4}\,dx+\frac{3}{2}\int\frac{\sec^2t}{(2\tan t)^2+4}\,dt\\[1ex] &=\frac{1}{2}\int\frac{du}{u}+\frac{3}{2}\int\frac{\sec^2t}{(2\tan t)^2+4}\,dt \end{align*}$$ The first integral should be obvious. The second will be, too, after some simplification.