I was wondering how to express the closed-form solution of the following double line integral in $\mathbb{R}^n$:
$$ \int_{\mathbf{x}_1}^{\mathbf{x}_2} \int_{\mathbf{y}_1}^{\mathbf{y}_2} \frac{1}{|\mathbf{x} - \mathbf{y}|} d\mathbf{x} d\mathbf{y} $$
where
$$ \mathbf{x} = [x_1, x_2, \dots, x_n]^T $$ $$ \mathbf{y} = [y_1, y_2, \dots, y_n]^T $$ $$ |\mathbf{x}| = \sqrt{\sum_{k=1}^n|x_k|^2} $$ $$ |\mathbf{y}| = \sqrt{\sum_{k=1}^n|y_k|^2} $$
given that $\mathbf{x}_1, \mathbf{x}_2, \mathbf{y}_1, \mathbf{y}_2$ are all distinct from each other and the lines $\mathbf{x}_1, \mathbf{x}_2$ and $\mathbf{y}_1, \mathbf{y}_2$ do not cross each other. That is, it is granted that $|\mathbf{x} - \mathbf{y}| > 0$ when evaluating the integral.
I thought on using coordinate translation to do the integration along the line $\mathbf{z}_1, \mathbf{z}_2$ in relation to the origin
$$ \int_{\mathbf{z}_1}^{\mathbf{z}_2} \frac{1}{|\mathbf{z}|} d\mathbf{z} = \ln{\frac{ \sum_{k=1}^n \mathbf{z}_{2,k}(\mathbf{z}_{2,k} - \mathbf{z}_{1,k}) + \sqrt{ (\sum_{k=1}^n \mathbf{z}_{2,k}) (\sum_{k=1}^n (\mathbf{z}_{2,k} - \mathbf{z}_{1,k})) } }{ \sum_{k=1}^n \mathbf{z}_{1,k}(\mathbf{z}_{2,k} - \mathbf{z}_{1,k}) + \sqrt{ (\sum_{k=1}^n \mathbf{z}_{1,k}) (\sum_{k=1}^n (\mathbf{z}_{2,k} - \mathbf{z}_{1,k})) } }} $$
where $\mathbf{z}_{1,k}$ is the $k$-th coordinate of the point $\mathbf{z}_{1}$.
But then I run out of ideias on how to proceed further and do the second integral taking advantage of that result...
Since you have a double line integral, you cam try making that explicit $\int_{x_{1}}^{x_{2}}dxF(x)=|x_{1}-x_{2}|\int_{0}^{1}dtF(x_{1}+t(x_{2}-x_{1}))$
thus
$\int_{x_{1}}^{x_{2}}\int_{y_{1}}^{y_{2}}dxdyF(|x-y|)=|x_{1}-x_{2}||y_{1}-y_{2}|\int_{0}^{1}\int_{0}^{1}dsdtF(|x_{1}-y_{1}+t(x_{2}-x_{1})-s(y_{2}-y_{1})|)$
The interesting part of the integral becomes
$\int_{0}^{1}\int_{0}^{1}dsdtF(|x_{1}-y_{1}+t(x_{2}-x_{1})-s(y_{2}-y_{1})|)=\int_{0}^{1}\int_{0}^{1}dsdtF(\sqrt{A+Bt+Cs+Dts+Et^{2}+Fs^{2}})$
For the correct values of $A,B,C,D,E,F$ So you are left to solve
$\int_{0}^{1}\int_{0}^{1}dsdt {1\over\sqrt{A+Bt+Cs+Dts+Et^{2}+Fs^{2}}}$ which might be easier