What is the gravitational force imposed by a rectilinear 2d body? I'm starting with the simplest problem I can relate to mine: the force imposed on a point mass at the origin by a rectangle that is orthogonal to the $x$- and $y$-axis, stretching from $(x_1, y_1)$ to $(x_2, y_2)$.
I'm using Newton's formula, but currently ignoring the mass and density.
I start off doing something like 

$\displaystyle\int\nolimits^{y_2}_{y_1}\int\nolimits^{x_2}_{x_1}\frac{1}{x^2 + y^2}dxdy$

Where $x^2+y^2$ equals the squared distance between $(x,y)$ and the origin.
This leads me to the following integral: 

$\displaystyle\int^{y_2}_{y_1}(\frac{1}{y}\arctan\frac{x_2}{y}) - (\frac{1}{y}\arctan\frac{x_1}{y})dy$

Is there a better way? I can't find any way to solve this.

As far as I can see, the final solution takes the form of:
(based on Christian Blatters answer)
$$F_1=-ln({c+\sqrt{a^2+c^2})}+ln({d+\sqrt{a^2+d^2})}+ln({c+\sqrt{b^2+c^2})}-ln({d+\sqrt{b^2+d^2})} \ .$$
