<<PolylogTools`

In1[n_Integer /; n ≥ 1, x_,y_]:=G@@Join[{y} , Table[0, {n-1}], {x, 1}]
In1I[n_Integer /; n ≥ 1, x_,y_]:=Integrate[-PolyLog[n, t/x] / (t - y), {t, 0, 1}]

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

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

wt6 = ((4 (π^6))/315)+G[0,0,0,0,0,(1/x),1]-G[0,0,0,0,0,(1/y),1]+5 G[0,0,0,0,0,(y/x),1]-G[(1/y),0,0,0,0,(1/x),1]+G[y,0,0,0,0,x,1]+G[0,0,0,0,(1/y),1] Log[x]+(1/2) Li[{4},{y}] ((Log[x])^2)-(1/6) Li[{3},{y}] ((Log[x])^3)+(1/24) Li[{2},{y}] ((Log[x])^4)-(1/120) Li[{1},{y}] ((Log[x])^5)+I π (-G[0,0,0,0,(1/y),1]-Li[{4},{y}] Log[x]+(1/2) Li[{3},{y}] ((Log[x])^2)-(1/6) Li[{2},{y}] ((Log[x])^3)+(1/24) Li[{1},{y}] ((Log[x])^4)+(((Log[x])^5)/120))-G[0,0,0,0,(y/x),1] Log[(x/y)]+(1/90) (π^4) (-2 Li[{2},{y}]+2 Li[{1},{y}] Log[x]+((Log[x])^2)+3 ((Log[(x/y)])^2))+(1/720) (-((Log[x])^6)+((Log[(x/y)])^6))+(1/3) (π^2) (-Li[{4},{y}]+Li[{3},{y}] Log[x]-(1/2) Li[{2},{y}] ((Log[x])^2)+(1/6) Li[{1},{y}] ((Log[x])^3)+(1/24) (((Log[x])^4)+((Log[(x/y)])^4)))/.{G[0,0,0,0,0,x_,1]:>-Li[{6}, {1/x}],G[0,0,0,0,x_,1]:>-Li[{5}, {1/x}]};

wt5 = 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]/.{PolyLog[n_, x_]:>Li[{n}, {x}],Log[1-y]->-Li[{1},{y}],G[0,0,0,0,x_,1]:>-Li[{5}, {1/x}]};

wt4 =  ((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)));

wt3 =  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));

wt2 = ((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));

(* ---------------- *)
f[alpha_Integer /; alpha ≥ 0, beta_ /; beta ≥ -1, x_, y_] := -(-1)^alpha/alpha! (beta Log[x/y]^alpha + Log[x]^alpha) + LiM[alpha, x, y];

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

test6 = In1[5,x,y] - In1[5,1/x,1/y] +
 -(Li[{6}, {x}] + 5 Li[{6}, {x/y}] - Log[x/y] Li[{5}, {x/y}]) +
(I Pi) f[5, 0, x, y] +
-(2 Zeta[0]) f[6, -1, x, y]+
-(2 Zeta[2]) f[4, 1, x, y]+
-(2 Zeta[4]) f[2, 3, x, y]+
-(2 Zeta[6]) f[0, 5, x, y];

wt6 - test6//Expand

test4 = In1[3,x,y] - In1[3,1/x,1/y] +
 -(Li[{4}, {x}] + 3 Li[{4}, {x/y}] - Log[x/y] Li[{3}, {x/y}]) +
(I Pi) f[3, 0, x, y] +
-(2 Zeta[0]) f[4, -1, x, y]+
-(2 Zeta[2]) f[2, 1, x, y]+
-(2 Zeta[4]) f[0, 3, x, y];

test4 - wt4//Expand

test2 = In1[1,x,y] - In1[1,1/x,1/y] +
 -(Li[{2}, {x}] +  Li[{2}, {x/y}] - Log[x/y] Li[{1}, {x/y}]) +
(I Pi) f[1,0, x, y] +
-(2 Zeta[0]) f[2,-1, x, y]+
-(2 Zeta[2]) f[0,1, x, y];

test2 - wt2//Expand

test5 = In1[4,x,y] + In1[4,1/x,1/y] +
 +(Li[{5}, {x}] + 4 Li[{5}, {x/y}] - Log[x/y] Li[{4}, {x/y}]) +
(I Pi) f[4, 0, x, y] +
-(2 Zeta[0]) f[5, -1, x, y]+
-(2 Zeta[2]) f[3, 1, x, y]+
-(2 Zeta[4]) f[1, 3, x, y];

test5 - wt5//Expand

test3 = In1[2,x,y] + In1[2,1/x,1/y] +
 +(Li[{3}, {x}] + 2 Li[{3}, {x/y}] - Log[x/y] Li[{2}, {x/y}]) +
(I Pi) f[2, 0, x, y] +
-(2 Zeta[0]) f[3, -1, x, y]+
-(2 Zeta[2]) f[1, 1, x, y];

test3 - wt3//Expand

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

eqn8 = In1I[7,x,y] - In1I[7,1/x,1/y] +
 -(Li[{8}, {x}] + 7Li[{8}, {x/y}] - Log[x/y] Li[{7}, {x/y}]) +
(I Pi) f[7, 0, x, y] +
-(2 Zeta[0]) f[8, -1, x, y]+
-(2 Zeta[2]) f[6, 1, x, y]+
-(2 Zeta[4]) f[4, 3, x, y]+
-(2 Zeta[6]) f[2, 5, x, y]+
-(2 Zeta[8]) f[0, 7, x, y];

test8 = eqn8/.Li[{n_}, {x_}]:>PolyLog[n,x]/.{x->-6-52/100,y->-9-882/1000}/.In1->In1I

N[test8, 50]

eqn7 = In1I[6,x,y] + In1I[6,1/x,1/y] +
 +(Li[{7}, {x}] + 6Li[{7}, {x/y}] - Log[x/y] Li[{6}, {x/y}]) +
(I Pi) f[6, 0, x, y] +
-(2 Zeta[0]) f[7, -1, x, y]+
-(2 Zeta[2]) f[5, 1, x, y]+
-(2 Zeta[4]) f[3, 3, x, y]+
-(2 Zeta[6]) f[1, 5, x, y];

test7 = eqn7/.Li[{n_}, {x_}]:>PolyLog[n,x]/.{x->-6-52/100,y->-9-882/1000}/.In1->In1I

N[test7, 50]

generalrel[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}] - Log[x/y] Li[{n}, {x/y}]) +
I Pi f[n, 0, x, y] +
-2 Sum[Zeta[j] f[n+1  -  j, j-1, x, y], {j, 0, n+1, 2}]

generalrelI[n_Integer/; n≥1] := In1I[n,x,y] + (-1)^n In1I[n,1/x,1/y] +
(-1)^n (Li[{n+1}, {x}] + n Li[{n+1}, {x/y}] - Log[x/y] Li[{n}, {x/y}]) +
I Pi f[n, 0, x, y] +
-2 Sum[Zeta[j] f[n+1  -  j, j-1, x, y], {j, 0, n+1, 2}]

generalrel[5] - wt6//Expand

generalrel[4] - wt5//Expand

generalrel[3] - wt4//Expand

generalrel[2] - wt3//Expand

generalrel[1] - wt2//Expand

generalrelI[6] - eqn7//Expand

generalrelI[7] - eqn8//Expand

N[generalrelI[12]/.Li[{n_}, {x_}]:>PolyLog[n,x]/.{x->-6-52/100,y->-9-882/1000},50]

N[generalrelI[33]/.Li[{n_}, {x_}]:>PolyLog[n,x]/.{x->-6-52/100,y->-9-882/1000},50]

N[generalrelI[33]/.Li[{n_}, {x_}]:>PolyLog[n,x]/.{x->-6-52/100 + 435I/234,y->-9-882/1000 + 543I/231},50]

N[generalrelI[33]/.Li[{n_}, {x_}]:>PolyLog[n,x]/.{x->-6 + 3I,y->-9 + 2I},50]

N[generalrelI[123]/.Li[{n_}, {x_}]:>PolyLog[n,x]/.{x->-17 - 25/100,y->-39-76/100},50]

N[In1I[n,x,y] + (-1)^n In1I[n,1/x,1/y]/.{n->123,x->-17 - 25/100,y->-39-76/100}, 50]