<<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)!

identity[5]

CreateTensor[identity[1]]

CreateTensor[identity[2]]

CreateTensor[identity[3]]

CreateTensor[identity[4]]

CreateTensor[identity[5]]

(* Claude has confirmed that identity 7 and identity 8 both have symbol 0 too *)