<<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[3]

In1[5,x,y] - In1[5,1/x,1/y] - (-(1)Li[{6}, {x}] - Li[{6}, {y}] + 5 Li[{6}, {x/y}])//CreateTensor//sh//flatTensor//Expand

lhs[n_Integer /; n ≥ 1] := In1[n, x, y] + (-1)^n In1[n, 1/x, 1/y]

rhs[n_Integer /; n ≥ 1] := -(-1)^n (Li[{n+1}, {x}] + n Li[{n+1}, {x/y}]) - Li[{n+1}, {y}]

lhs[5] - rhs[5]//CreateTensor//sh//flatTensor//Expand

lhs[4] - rhs[4]//CreateTensor//sh//flatTensor//Expand

lhs[3] - rhs[3]//CreateTensor//sh//flatTensor//Expand

lhs[2] - rhs[2]//CreateTensor//sh//flatTensor//Expand

lhs[1] - rhs[1]//CreateTensor//sh//flatTensor//Expand