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