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Let $W$ be a rational function of $8$ variables $a,b,c,d,e,f,g,h$ from this file, e.g.:

W = (256*b*d*f*(a^5 + b^4*(c - 2*e) - a^4*(3*c + 2*e) - c*(c^2 + d^2)*(e^2 + f^2) + a^3*(2*b^2 + 3*c^2 + d^2 + 6*c*e + e^2 + f^2) + b^2*(c^3 - 2*c^2*e + 2*d^2*e + c*(d^2 - e^2 - f^2)) + a*(b^4 + 2*c^3*e + 2*c*d^2*e + 3*c^2*(e^2 + f^2) + d^2*(e^2 + f^2) + b^2*(-c^2 - 3*d^2 + 6*c*e + e^2 + f^2)) - a^2*(c^3 + 6*c^2*e + 2*d^2*e + 2*b^2*(c + 2*e) + c*(d^2 + 3*(e^2 + f^2))))*(e - g)*h* (c^5 + d^4*(e - 2*g) - c^4*(3*e + 2*g) - e*(e^2 + f^2)*(g^2 + h^2) + c^3*(2*d^2 + 3*e^2 + f^2 + 6*e*g + g^2 + h^2) + d^2*(e^3 - 2*e^2*g + 2*f^2*g + e*(f^2 - g^2 - h^2)) + c*(d^4 + 2*e^3*g + 2*e*f^2*g + 3*e^2*(g^2 + h^2) + f^2*(g^2 + h^2) + d^2*(-e^2 - 3*f^2 + 6*e*g + g^2 + h^2)) - c^2*(e^3 + 6*e^2*g + 2*f^2*g + 2*d^2*(e + 2*g) + e*(f^2 + 3*(g^2 + h^2)))))/ ((a^2 + b^2)*((-1 + c)^2 + d^2)*((b + d)^2 + (a - c)^2)*((b - d)^2 + (a - c)^2)*((b + f)^2 + (a - e)^2)*((b - f)^2 + (a - e)^2)*((d + f)^2 + (c - e)^2)*((d - f)^2 + (c - e)^2)*((d + h)^2 + (c - g)^2)*((d - h)^2 + (c - g)^2)*((f + h)^2 + (e - g)^2)*((f - h)^2 + (e - g)^2))

I would like to integrate this over $\mathbb{H}^4$ (where $\mathbb{H} = \mathbb{R} \times \mathbb{R}_{>0}$) and divide by $(2\pi)^8$. Naively:

Integrate[W, {a, -Infinity, Infinity}, {b, 0, Infinity}, {c, -Infinity, Infinity}, {d, 0, Infinity}, {e, -Infinity, Infinity}, {f, 0, Infinity}, {g, -Infinity, Infinity}, {h, 0, Infinity}]/(2 Pi)^8

Of course, this does not give me an answer in reasonable time. How can I effectively integrate this function with Mathematica, either symbolically or numerically?

One strategy for numerical evaluation in Mathematica which works for some of the integrands (but not all) is listed in Implementation 17 in Appendix A of arXiv:1702.00681 [math.CO]. Ideally, an answer to this question would give an algorithm that works uniformly for each of the integrands.

These integrals are weights of Kontsevich graphs. See also my question on Math.SE.