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This is an approach for a numerical solution using mpmath:

sage: x, y = var('x, y')
sage: h = 1/(log(abs(x^4 + 13*x^3*y - 30*x^2*y^2 + 2*x*y^3 - 3*x^2*y + x*y^2 + x^2 + x*y + y^2 + 2*x + 13))^log(abs(x^4 + 13*x^3*y - 30*x^2*y^2 + 2*x*y^3 - 3*x^2*y + x*y^2 + x^2 + x*y + y^2 + 2*x + 13)))
sage: from sage.ext.fast_callable import fast_callable
sage: hh = fast_callable(h, vars=[x, y])
sage: import mpmath as mp
sage: print(mp.quad(hh, [1, 5], [1, 5]))
(124.078805233118 + 10.8586740662171j)

A good reference on integration, with many examples, is Ch. 14 of Calcul Mathematique avec Sage (link to English translation).