`int 1 / sqrt(-x^2-4x) dx` Find or evaluate the integral by completing the square

By completing the square and making simple substitution, we will reduce this integral to a table one.

`-x^2-4x = -(x^2 + 4x + 4) + 4 = -(x+2)^2 + 4 = 4 - (x+2)^2.`

Now make a substitution `y = (x+2)/2,` then `dy = dx/2,` `dx = 2 dy`  and integral becomes

`int 1/sqrt(4-4y^2)*2 dy = int (dy)/sqrt(1-y^2) = arcsin(y) + C = arcsin((x+2)/2) + C,`

where `C` is any constant.