You can do this without integration, by performing the following steps.
Calculate the coordinates of the vertices of your rectangle. Let the vertices be $v_1,v_2,v_3,v_4$.
Let your point "source" be $O$. Normalize the vectors $Ov_j$ by dividing each vector on its length. This gives you 4 points on the unit sphere centered at $O$.
Connect those 4 points by great circles in the correct order. You obtain a spherical quadrilateral. Find its interior angles, by spherical trigonometry. (They are the same as dihedral angles of the cone built on $Ov_j$.
Your solid angle is the area of this spherical quadrilateral, and once you know the interior angles it is equal to the sum of these angles minus $2\pi$.