Derive a holonomic system for the integrand in Example 1

<< RISC`HolonomicFunctions` (* the pacakge is available in http://www.risc.jku.at/research/combinat/software/ergosum/installation.html#download *)

HolonomicFunctions Package version 1.7.3 (21-Mar-2017)
written by Christoph Koutschan
Copyright Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

stheta=(2 s)/(1+s^2); (* make a universal transformation for Sin[theta] *)

ctheta=(1-s^2)/(1+s^2); (* make a universal transformation for Cos[theta] *)

sphi=(2 t)/(1+t^2); (* make a universal transformation for Sin[phi] *)

cphi=(1-t^2)/(1+t^2);(* make a universal transformation for Cos[phi] *)

R=s1 (b sphi stheta+d cphi ctheta-m11)^2+s2 (-b sphi ctheta+d cphi stheta-m21)^2+s2 (b cphi ctheta+d sphi stheta-m22)^2+s1 (-b cphi stheta+d sphi ctheta)^2; (* input R in the integrand *)

f=2/(1+t^2 )2/(1+s^2) (d^2-b^2) (s1 s2)/(2 Pi)^2 Exp[-1/2 R]; (*input the integral *)

Timing[ann=Annihilator[f,{Der[d],Der[b],Der[s],Der[t]}];](* compute a holonomic system ann of f *)

ann >> ec1_ann.m;