Answer
The integral is: $$-\frac{4}{3}\arctan\frac{x+2}{3} + \frac{1}{2} \ln(x^2+4x+13)+c$$ where $c$ is arbitratry.
The graph of the antiderivatives is on the following figure.
The graphs are shifted by a constant.
Work Step by Step
Using Wolfram Mathematica (which is an example of computer algebra system) with the code
t = Integrate [(x - 2)/(x^2 + 4 x + 13), x]
Plot [{t + 2, t - 5}, {x, -10, 10}]
We get that integral $$-\frac{4}{3}\arctan\frac{x+2}{3} + \frac{1}{2} \ln(x^2+4x+13)+c.$$
We get two antiderivatives for two different choices for $c$. We have chosen $c=2$ (blue) and $c=-5$ (orange) and they are plotted on the graph. The graphs are shifted by a constant.
