$$\int_c^d\int_a^b\ln|x-y|\ dx\ dy$$
Here we have double integral of a logarithmic function. I really worked hard but nothing worked out. Any kind of insight, approximation or even document suggestion towards solution of this integral equation will be much appreciated. It's really important part of my electromagnetics exam. Thanks in advance.
HINT:
There are six cases; (i) $a<c<b<d$, (ii) $c<a<b<d$, (iii) $a<c<d<b$, (iv) $c<a<d<b$, (v) $a<b<c<d$, and (vi) $c<d<a<b$.
To facilitate, draw a diagram of the rectangle $[a,b]\times [c,d]$ for each of the four cases and the line $y=x$.
Be careful with respect to the relative order of $a$, $b$, $c$, and $d$ when removing the absolute value signs with $\pm (x-y)$.
For example, in Case (i) for which $a<c<b<d$, we can write
$$\begin{align} \int_c^d \int_a^b \log|x-y|\,dx\,dy&=\int_b^d \int_a^b \log(y-x)\,dx\,dy\\\\ &+\int_c^b\int_a^y \log(y-x)\,dx\,dy+\int_a^b\int_y^b \log(x-y)\,dx\,dy \end{align}$$
