<<PolylogTools`

In1[n_Integer /; n ≥ 1, x_,y_]:=G@@Join[{y} , Table[0, {n-1}], {x, 1}]

Li[{5}, {x_}] := -G[0, 0, 0, 0, 1/x, 1]
Li[{6}, {x_}] := -G[0, 0, 0, 0, 0, 1/x, 1]
LiM[n_, x_, y_] := Sum[(-Log[x])^i Li[{n-i},{y}]/i!,{i,0,n-1}]

identity[n_Integer /; n ≥1] := In1[n, x, y] + (-1)^n In1[n, 1/x, 1/y]+
(-1)^n (Li[{n+1},{x}]+n Li[{n+1},{x/y}])+
LiM[n+1, x, y] - 
(-1)^n Log[x/y] Li[{n}, {x/y}] -
(-1)^n(Log[x/y]^(n+1) - Log[x]^(n+1))/(n+1)!

(* --------- *)

GI[a3_,a2_,a1_,1] := Integrate[1/(x1 - a1) 1/(x2-a2) 1/(x3 - a3), {x3,0,1},{x2,0,x3},{x1,0,x2}]
GN[a3_,a2_,a1_,1] := NIntegrate[1/(x1 - a1) 1/(x2-a2) 1/(x3 - a3), {x3,0,1},{x2,0,x3},{x1,0,x2}]

(* --------- *)

tolog[expr_] := (((expr)/.{G[0, a_, b_]:>-Li[{2}, {b/a}],G[a_,b_]:>GLog[a,b]}))/.Li[{n_}, {x_}]:>PolyLog[n,x]

expandlog[expr_] := ((expr)/.Log[a_]:>Log[Factor[a]]/.Log[a_Times]:> (Plus@@(Log/@(List@@a)))/.Log[a_^b_]:>b Log[a])

canonicallog[expr_] := ((expr)/.{Log[-1+x]:>(I Pi + Log[1-x]),Log[-1+y]:>(I Pi + Log[1-y]),Log[-x+y]:>(I Pi + Log[x-y])})

(* --------- *)

id = identity[2]

delta1 = Delta111[id]//tolog//expandlog//canonicallog

(delta1/(-I)//Expand)/.{CT[Pi, rest__]:>CT[rest]}//ToSymbol

CreateTensor[-Log[x]^2/2! + LiM[2, x, y]]

delta1corr = I Pi(-Log[x]^2/2! + LiM[2, x, y]);

id1 = identity[2] + delta1corr

Delta111[id1]//tolog//expandlog//canonicallog

(* -------------- *)

i11rel = ((2(π^2))/3)-G[(1/y),(1/x),1]+G[y,x,1]-Li[{2},{x}]-Li[{2},{(x/y)}]+Li[{2},{y}]-Li[{1},{y}] Log[x]+I π (Li[{1},{y}]+Log[x])+Li[{1},{(x/y)}] Log[(x/y)]+(1/2) (-((Log[x])^2)+((Log[(x/y)])^2))//tolog;

li2inv = PolyLog[2, x] + PolyLog[2, 1/x] - (- Pi^2/6- Log[-1/x]^2/2)//tolog;

delta2 = Delta21[id1]//tolog

delta2 = (delta2/.{G[1/y, 1/x, 1]->G[1/y,1/x,1]+i11rel,PolyLog[2,1/x]->PolyLog[2,1/x] - li2inv})//expandlog//canonicallog//ExpandAll

-6*delta2/.{Complex[0,_]->0}/.{CT[Pi^2, x__]:>CT[x]}//ToSymbol

CreateTensor[-2(Log[x/y] + Log[x] + LiM[1,x,y])]

delta2corr = (Pi^2/6)(-2(Log[x/y] + Log[x] + LiM[1,x,y]));

id2 = identity[2] + delta1corr + delta2corr

(id2//tolog)/.{x->-12,y->-19,G->GI}

N[%,20]

idfull = identity[2] + delta1corr + delta2corr + 0