# Integrating rational functions of several variables over $\mathbb{H}^4$

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.

• Not sure if it will help but you might want to have a look at this ArXiV article. Mar 9 '17 at 16:06
• A very good question, unfortunately underestimated. I guess you could elaborate a bit on the weights of Kontsevich graphs etc. Acting with Apart on W one can realize why a direct Integrate approach doesn't seem to be successful. Do you know any symmetries of the integrand? I believe it would be more resonable first to ask if one could improve implementation 17 from the linked paper to deal with a larger class of the integrands. Then some progress would be more definite. Mar 10 '17 at 9:08