Perhaps the simplest approach first computes the determinant of an arbitrary Cauchy matrix, here is a <A HREF="https://proofwiki.org/wiki/Value_of_Cauchy_Determinant">derivation</A>. Since every submatrix of a Cauchy matrix is also a Cauchy matrix, you can then find the inverse $H$ from the adjoint of $H$. 

Some pointers to worked-out calculations:

- <A HREF="https://www.scirp.org/pdf/6-3.2.pdf">A New Approach to Inversion of a Cauchy Matrix</A>

- <A HREF="https://proofwiki.org/wiki/Inverse_of_Cauchy_Matrix">Inverse of Cauchy Matrix</A>


For a direct evaluation of the sum of all entries of $H^{-1}$, see <A HREF="https://arxiv.org/abs/2301.09777">The entry sum of the inverse Cauchy matrix</A>