<<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[a6_, a5_, a4_, a3_,a2_,a1_,1] := Integrate[1/(x1 - a1) 1/(x2-a2) 1/(x3 - a3) 1/(x4-a4) 1/(x5-a5) 1/(x6-a6), {x6,0,1},{x5,0,x6}, {x4,0,x5}, {x3,0,x4},{x2,0,x3},{x1,0,x2}]
GN[a6_, a5_, a4_, a3_,a2_,a1_,1] := NIntegrate[1/(x1 - a1) 1/(x2-a2) 1/(x3 - a3) 1/(x4-a4) 1/(x5-a5) 1/(x6-a6), {x6,0,1},{x5,0,x6}, {x4,0,x5},{x3,0,x4},{x2,0,x3},{x1,0,x2}]

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

tolog[expr_] := (((expr)/.{G[0, 0,0, 0,0,a_, b_]:>-PolyLog[6, b/a],G[0,0, 0,0,a_, b_]:>-PolyLog[5, b/a],G[0, 0,0,a_, b_]:>-Li[{4}, {b/a}],G[0, 0,a_, b_]:>-Li[{3}, {b/a}],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[5]

delta1 = Delta111111[id]//tolog//expandlog//canonicallog//Expand

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

CreateTensor[Log[x]^5/5! + LiM[5, x, y]]

delta1corr = I Pi(Log[x]^5/5! + LiM[5, x, y])

id1 = identity[5] + delta1corr

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

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 = Delta21111[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]^4 + Log[x]^4)/4! - LiM[4,x,y])]

delta2corr = (Pi^2/6)(2((Log[x/y]^4 + Log[x]^4)/4! - LiM[4,x,y]))

id2 = identity[5] + delta1corr + delta2corr

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

i21rel = G[(1/y),0,(1/x),1]+G[y,0,x,1]+Li[{3},{x}]+2 Li[{3},{(x/y)}]+Li[{3},{y}]-Li[{2},{y}] Log[x]+(1/2) Li[{1},{y}] ((Log[x])^2)+I π (Li[{2},{y}]-Li[{1},{y}] Log[x]-(((Log[x])^2)/2))-Li[{2},{(x/y)}] Log[(x/y)]-(1/3) (π^2) (Li[{1},{y}]+Log[x]+Log[(x/y)])+(1/6) (((Log[x])^3)-((Log[(x/y)])^3))//tolog;

li3inv = PolyLog[3,1/x] - PolyLog[3,x] - Log[x]^3/3! + I Pi Log[x]^2/2! + Pi^2/3 Log[x]//tolog;

delta3 = Delta3111[id2]//tolog

(delta3/.{G[1/y,0,1/x,1]->G[1/y,0,1/x,1]-i21rel,PolyLog[3,1/x]->PolyLog[3,1/x]-li3inv})//expandlog//canonicallog//ExpandAll

delta3corr = 0

id3 = identity[5] + delta1corr + delta2corr + delta3corr

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

i31rel = ((4 (π^4))/45)-G[(1/y),0,0,(1/x),1]+G[y,0,0,x,1]-Li[{4},{x}]-3 Li[{4},{(x/y)}]+Li[{4},{y}]-Li[{3},{y}] Log[x]+(1/2) Li[{2},{y}] ((Log[x])^2)-(1/6) Li[{1},{y}] ((Log[x])^3)+I π (Li[{3},{y}]-Li[{2},{y}] Log[x]+(1/2) Li[{1},{y}] ((Log[x])^2)+(((Log[x])^3)/6))+Li[{3},{(x/y)}] Log[(x/y)]+(1/24) (-((Log[x])^4)+((Log[(x/y)])^4))+(1/3) (π^2) (-Li[{2},{y}]+Li[{1},{y}] Log[x]+(1/2) (((Log[x])^2)+((Log[(x/y)])^2)))//tolog;

li4inv = PolyLog[4,x] + PolyLog[4,1/x] + (2Pi I)^4/4! BernoulliB[4,1/2 - Log[-1/x]/(2Pi I)]//expandlog//Expand

delta4 = Delta411[id3]//tolog

delta4 = (delta4/.{G[1/y,0,0,1/x,1]->G[1/y,0,0,1/x,1]+i31rel,PolyLog[4,1/x]->PolyLog[4,1/x]-li4inv})//expandlog//canonicallog//ExpandAll

-90*delta4/.{Complex[0,_]->0}/.{CT[Pi^4, x__]:>CT[x]}//ToSymbol

CreateTensor[2Log[x]^2/2! + 6Log[x/y]^2/2!+2Log[x]Li[{1}, {y}] - 2Li[{2},{y}]]

delta4corr = (Pi^4/90)(2Log[x]^2/2! + 6Log[x/y]^2/2!+2Log[x]Li[{1}, {y}] - 2Li[{2},{y}])

id4 = identity[5] + delta1corr + delta2corr + delta3corr + delta4corr

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

i41rel = G[(1/y),0,0,0,(1/x),1]+G[y,0,0,0,x,1]-(1/24) ((Log[x])^4) Log[1-y]+(1/120) (((Log[x])^5)-((Log[(x/y)])^5))+(1/90) (π^4) (2 Log[1-y]-8 Log[(x/y)]-2 Log[y])-(1/6) ((Log[x])^3) PolyLog[2,y]+(1/2) ((Log[x])^2) PolyLog[3,y]-(1/3) (π^2) (-(1/2) ((Log[x])^2) Log[1-y]+(1/6) (((Log[x])^3)+((Log[(x/y)])^3))-Log[x] PolyLog[2,y]+PolyLog[3,y])-Log[(x/y)] PolyLog[4,(x/y)]-Log[x] PolyLog[4,y]+I π (-(1/24) ((Log[x])^4)+(1/6) ((Log[x])^3) Log[1-y]+(1/2) ((Log[x])^2) PolyLog[2,y]-Log[x] PolyLog[3,y]+PolyLog[4,y])+PolyLog[5,x]+4 PolyLog[5,(x/y)]+PolyLog[5,y]//tolog;

li5inv = PolyLog[5,x] - PolyLog[5,1/x] + (2Pi I)^5/5! BernoulliB[5,1/2 - Log[-1/x]/(2Pi I)]//expandlog//Expand

delta5 = Delta51[id4]//tolog

(delta5/.{G[1/y,0,0,0,1/x,1]->G[1/y,0,0,0,1/x,1]-i41rel,PolyLog[5,1/x]->PolyLog[5,1/x]+li5inv})//expandlog//canonicallog//ExpandAll

delta5corr = 0

id5 = identity[5] + delta1corr + delta2corr + delta3corr + delta4corr + delta5corr

(id5//tolog)/.{x->-12,y->-15,G->GI}

N[%, 50]

N[-12.2081167438133896765742151574910463348218098/Zeta[6], 50]

12Zeta[6]

N[%, 20]

idfull = identity[5] + delta1corr + delta2corr + delta3corr + delta4corr + delta5corr+((4 (π^6))/315)

(idfull//tolog)/.{x->-16,y->-51,G->GI}

N[%,50]