<<PolylogTools`

GatherCiTiList[expr_]:=Block[{temp=flatTensor[Expand[expr]]},temp=GatherBy[List@@CollectCiTi[temp,1],Last];
temp=CleanUp/@temp;
Return[temp];];
CleanUp[list_List]:={list[[1,2]],Total[list[[All,1]]]}
CollectCiTi[sum_Plus,a_]:=CollectCiTi[#,a]&/@sum;
CollectCiTi[a_. ct_CiTi,b_]:=CollectCiTi[a,ct b];
CollectCiTi[sum_Plus,a_]:=CollectCiTi[#,a]&/@sum;
CollectCiTi[a_. ct_CiTi,b_]:=CollectCiTi[a,ct b];
CleanUp[list_List]:={list[[1,2]],Total[list[[All,1]]]}
GatherCiTiList[expr_]:=Block[{temp=flatTensor[Expand[expr]]},temp=GatherBy[List@@CollectCiTi[temp,1],Last];
temp=CleanUp/@temp;
Return[temp];];

basgen[i_,n_] := Table[If[i==k,1,0], {k, 1, n}];

I41[x_,y_] := G[y, 0, 0,0, x, 1];
I32[x_,y_] := G[0,y, 0, 0,x, 1];
I311[x_,y_,z_] := G[z,y, 0, 0,x, 1];

N5[x_]:=G[0,0,0,1,1,x]

L1[x_] := -G[1/x, 1];
L2[x_] := -G[0, 1/x, 1];
L3[x_] := -G[0, 0,1/x, 1];
L4[x_] := -G[0, 0, 0,1/x, 1];
L5[x_] := -G[0,0,0,0, 1/x, 1];

cr[{a_,b_,c_,d_}]:=((a-c)/(a-d))/((b-c)/(b-d))
cr[{a_,b_,c_}]:=((a-c)/(b-c))

cr[{Inf,b_,c_,d_}]:=cr[{d,c,b}]
cr[{a_,Inf,c_,d_}]:=cr[{c,d,a}]
cr[{a_,b_,Inf,d_}]:=cr[{b,a,d}]
cr[{a_,b_,c_,Inf}]:=cr[{a,b,c}]

cr[{a_,b_,c_,d_,e_}] := Sequence[cr[{a,b,c,d}],cr[{a,b,c,e}]];

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

I41[x,y] + (1/3)(I32[x,y] + I32[y,x])//CreateTensor//del//Expand

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

I32[x,y] + I32[1/x,1/y] - (6L5[x/y]+ 4L5[y] - L5[x])//CreateTensor//sh//Expand

I32[cr[{a,b,c,d,e}]] + I32[cr[{b,a,c,d,e}]]//CreateTensor//del//Expand

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

I32[cr[{a,b,c,d,e}]]+ I32[cr[{a,b,d,c,e}]] - I32[cr[{a,b,c,e,d}]] - I32[cr[{a,b,e,c,d}]] - (-6 L5[cr[{a,b,c,d}]] + 6 L5[cr[{a,b,c,e}]])//CreateTensor//sh//flatTensor//Expand

I32[(1/x),(y/x)]+I32[x,y]-I32[(1/y),(x/y)]-I32[y,x]//CreateTensor//del//Expand

Hold[I32i[cr[{a,b,c,d,e}]]+ I32i[cr[{a,b,d,c,e}]] - I32i[cr[{a,b,c,e,d}]] - I32i[cr[{a,b,e,c,d}]]]/.{a->Inf,b->0,c->1,d->x,e->y}//Release

I32[(1/x),(y/x)]+I32[x,y]-I32[(1/y),(x/y)]-I32[y,x] - (-6 L5[x] + 6 L5[y])//CreateTensor//sh//flatTensor//Expand

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

(I32[cr[{a,b,c,d,e}]] + I32[cr[{b,c,a,d,e}]] + I32[cr[{c,a,b,d,e}]]) +
(I32[cr[{a,b,c,e,d}]] + I32[cr[{b,c,a,e,d}]] + I32[cr[{c,a,b,e,d}]])//CreateTensor//del//Expand

(I32[-(1/(-1+x)),-(1/(-1+y))]+I32[-((1-x)/x),-((1-y)/y)]+I32[x,y]+I32[-(1/(-1+y)),-(1/(-1+x))]+I32[-((1-y)/y),-((1-x)/x)]+I32[y,x])-(-3(-2 L5[x/y] -2 L5[(1-y)/(1-x)]  - 2L5[y(1-x)/(x(1-y))]+
2L5[x] -L5[1-1/x] +  I41[1,x] + 
2L5[y] -L5[1-1/y] +  I41[1,y]))//CreateTensor//sh//flatTensor//Expand

(I32[-(1/(-1+x)),-(1/(-1+y))]+I32[-((1-x)/x),-((1-y)/y)]+I32[x,y]+I32[-(1/(-1+y)),-(1/(-1+x))]+I32[-((1-y)/y),-((1-x)/x)]+I32[y,x])-(-3(-2 L5[x/y] -2 L5[(1-y)/(1-x)]  - 2L5[y(1-x)/(x(1-y))]+
-2L5[x] -L5[1-1/x] +  N5[x] + 
-2L5[y] -L5[1-1/y] +  N5[y]))//CreateTensor//sh//flatTensor//Expand

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

S[{a_,b_,c_,d_,e_}] := I32[cr[{a,b,c,d,e}]] + I32[cr[{b,c,a,d,e}]] + I32[cr[{c,a,b,d,e}]]

Hold[(S[{a,b,c,d,e}] + S[{b,c,d,e,a}] + S[{c,d,e,a,b}] + S[{d,e,a,b,c}] + S[{e,a,b,c,d}]) - 
(S[{a,c,e,b,d}] + S[{c,e,b,d,a}] + S[{e,b,d,a,c}] + S[{b,d,a,c,e}] + S[{d,a,c,e,b}])]/.{a->Inf,b->0,c->1,d->x,e->y}//Release//CreateTensor//del//flatTensor//Expand

Hold[(S[{a,b,c,d,e}] + S[{b,c,d,e,a}] + S[{c,d,e,a,b}] + S[{d,e,a,b,c}] + S[{e,a,b,c,d}]) - 
(S[{a,c,e,b,d}] + S[{c,e,b,d,a}] + S[{e,b,d,a,c}] + S[{b,d,a,c,e}] + S[{d,a,c,e,b}]) -(51 L5[cr[{a,b,c,d}]]-9 L5[cr[{a,b,c,e}]]+15 L5[cr[{a,b,d,e}]]+3 L5[cr[{a,c,b,d}]]+3 L5[cr[{a,c,b,e}]]-9 L5[cr[{a,c,d,e}]]-9 L5[cr[{a,d,b,c}]]-15 L5[cr[{a,d,b,e}]]+3 L5[cr[{a,d,c,e}]]-39 L5[cr[{a,e,b,c}]]+45 L5[cr[{a,e,b,d}]]-39 L5[cr[{a,e,c,d}]]+15 L5[cr[{b,c,d,e}]]-15 L5[cr[{b,d,c,e}]]+45 L5[cr[{b,e,c,d}]]+18 I41[1,cr[{a,b,c,d}]]-18 I41[1,cr[{a,e,b,c}]]+18 I41[1,cr[{a,e,b,d}]]-18 I41[1,cr[{a,e,c,d}]]+18 I41[1,cr[{b,e,c,d}]])]/.{a->Inf,b->0,c->1,d->x,e->y}//Release//CreateTensor//sh//flatTensor//Expand

Hold[(S[{a,b,c,d,e}] + S[{b,c,d,e,a}] + S[{c,d,e,a,b}] + S[{d,e,a,b,c}] + S[{e,a,b,c,d}]) - 
(S[{a,c,e,b,d}] + S[{c,e,b,d,a}] + S[{e,b,d,a,c}] + S[{b,d,a,c,e}] + S[{d,a,c,e,b}]) -(-21 L5[cr[{a,b,c,d}]]-9 L5[cr[{a,b,c,e}]]+15 L5[cr[{a,b,d,e}]]+3 L5[cr[{a,c,b,d}]]+3 L5[cr[{a,c,b,e}]]-9 L5[cr[{a,c,d,e}]]-9 L5[cr[{a,d,b,c}]]-15 L5[cr[{a,d,b,e}]]+3 L5[cr[{a,d,c,e}]]+33 L5[cr[{a,e,b,c}]]-27 L5[cr[{a,e,b,d}]]+33 L5[cr[{a,e,c,d}]]+15 L5[cr[{b,c,d,e}]]-15 L5[cr[{b,d,c,e}]]-27 L5[cr[{b,e,c,d}]]+18 N5[cr[{a,b,c,d}]]-18 N5[cr[{a,e,b,c}]]+18N5[cr[{a,e,b,d}]]-18 N5[cr[{a,e,c,d}]]+18 N5[cr[{b,e,c,d}]])]/.{a->Inf,b->0,c->1,d->x,e->y}//Release//CreateTensor//sh//flatTensor//Expand

45- 4*18

Hold[-21 L5[cr[{a,b,c,d}]]-9 L5[cr[{a,b,c,e}]]+15 L5[cr[{a,b,d,e}]]+3 L5[cr[{a,c,b,d}]]+3 L5[cr[{a,c,b,e}]]-9 L5[cr[{a,c,d,e}]]-9 L5[cr[{a,d,b,c}]]-15 L5[cr[{a,d,b,e}]]+3 L5[cr[{a,d,c,e}]]+33 L5[cr[{a,e,b,c}]]-27 L5[cr[{a,e,b,d}]]+33 L5[cr[{a,e,c,d}]]+15 L5[cr[{b,c,d,e}]]-15 L5[cr[{b,d,c,e}]]-27 L5[cr[{b,e,c,d}]]]/.{cr->cri,L5->L5x}//Release

%/3//Expand

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

(* ------- *)

wedge[a_Plus, b_] := wedge[#, b]&/@a
wedge[a_, b_Plus] := wedge[a,#]&/@b
wedge[lambda_?NumberQ * a_CiTi,b__] := lambda  wedge[a,b]
wedge[a___, mu_?NumberQ* b_CiTi] := mu wedge[a,b]
wedge[CiTi[a___], CiTi[b___]] := TensWedge[Tens@@ElTens/@{a},Tens@@ElTens/@{b}]

wedge[lambda_?NumberQ * a_TensWedge,b__] := lambda  wedge[a,b]
wedge[a___, mu_?NumberQ* b_TensWedge] := mu wedge[a,b]
wedge[CiTi[a___], TensWedge[b___]] := TensWedge[Tens@@ElTens/@{a},TensWedge[b]]

LiTensor[n_,x_] := -CiTi@@Join[{1-x},Table[x, {n-1}]]//ReduceTensorProduct//sh//flatTensor//Expand;

wedge[{x_,n_}, {y_,m_}] := wedge[LiTensor[n,x], LiTensor[m,y]]
wedge[{x_,n_}, {y_,m_}, {z_, o_}] := wedge[LiTensor[n,x], wedge[LiTensor[m,y], LiTensor[o,z]]]

I41m[x_,y_] := (I41[x,y] + I41[x,1/y])/2;
I41mi[x_,y_] := (I41i[x,y] + I41i[x,1/y])/2;

(CreateTensor[I41m[x,y]]//del//Expand)-(-wedge[{x,2},{y,3}])

(I41[x,y] //CreateTensor//del//Expand) - (-wedge[{x,2},{y,3}]  - wedge[{y,2},{x,3}])//Expand

(I32[x,y] //CreateTensor//del//Expand) - (2wedge[{x,2},{y,3}] + wedge[{y,2},{x,3}] - wedge[{y,2},{x/y,3}] + wedge[{x,2},{x/y,3}])//Expand

I32[x,y] - (-1/2)(3I41[x,y] + I41[x,1/y] + I41[x,x/y] + I41[x,y/x] - I41[y, x/y] - I41[y, y/x])//CreateTensor//del//Expand

i32as41 = I32[x,y] - (-1/2)(3I41[x,y] + I41[x,1/y] + I41[x,x/y] + I41[x,y/x] - I41[y, x/y] - I41[y, y/x])//CreateTensor//sh//Expand;

candidate = (1/3) (I32[x,y] - (-1/2)(3I41[x,y] + I41[x,1/y] + I41[x,x/y] + I41[x,y/x] - I41[y, x/y] - I41[y, y/x]))-2 L5[-(1/(-1+x))]+(19/12) L5[(1/x)]-L5[(x/(-1+x))]+L5[((-1+x)/(x-y))]+L5[(x/(x-y))]+L5[-(((-1+x) (x-y))/(x (((-1+y))^2)))]+(3/4) L5[((((x-y))^2)/(x (((-1+y))^2)))]+2 L5[-(1/(-1+y))]+(3/2) L5[-((-1+x)/(x (-1+y)))]-4 L5[-((x-y)/(x (-1+y)))]+(1/2) L5[-((((x-y))^2)/((-1+x) x (-1+y)))]-2 L5[((-1+y)/(-1+x))]+(3/32) L5[((((x-y))^2)/((((-1+y))^2) (y^2)))]-(1/18) L5[-((x (((-1+y))^3))/((((-1+x))^3) (y^2)))]+(1/12) L5[(1/y)]+(1/4) L5[(1/(x y))]-L5[((x-y)/((-1+x) y))]-(1/18) L5[-((((x-y))^3)/((((-1+x))^3) x y))]-(1/9) L5[((((x-y))^3)/(x (((-1+y))^3) y))]-(3/2) L5[-((x-y)/((-1+y) y))]+L5[-((((x-y))^2)/((-1+x) (-1+y) y))]+(1/4) L5[(y/(x^2))]+(1/4) L5[(y/x)]+2 L5[-(y/(x-y))]+(3/2) L5[-(((-1+x) y)/(x (x-y)))]-(1/9) L5[-(((((-1+x))^3) y)/((x^2) (((-1+y))^3)))]-(1/18) L5[(((((x-y))^3) y)/((x^2) (((-1+y))^3)))]+(3/4) L5[(((((-1+x))^2) y)/(x (((-1+y))^2)))]+(1/2) L5[-(((-1+x) (x-y) y)/(x (((-1+y))^2)))]+L5[(y/(-1+y))]-2 L5[(((-1+x) y)/(x (-1+y)))]-(5/12) L5[((y^2)/x)]-L5[((-1+y)/(-x+y))]+L5[-(((((-1+x))^2) y)/(x (-1+y) (-x+y)))]+(1/2) L5[-(((-1+y) (-x+y))/((((-1+x))^2) y))]+(3/4) L5[((((-x+y))^2)/((((-1+x))^2) y))]-(1/9) L5[((x (((-x+y))^3))/((((-1+x))^3) (y^2)))]-N5[-(1/(-1+x))]+N5[((-1+x)/(x-y))]+N5[-(1/(-1+y))]-N5[-((x-y)/(x (-1+y)))]+N5[-(y/(x-y))]//CreateTensor//sh//flatTensor//Expand

candidate2 = (22/9) (I32[x,y] - (-1/2)(3I41[x,y] + I41[x,1/y] + I41[x,x/y] + I41[x,y/x] - I41[y, x/y] - I41[y, y/x]))+(5/3) L5[-(1/(-1+x))]+(23/18) L5[(1/x)]-(20/3) L5[(x/(-1+x))]+(1/12) L5[(1/(((x-y))^2))]-(4/9) L5[(((-1+x) x)/(((x-y))^2))]+(4/3) L5[-(1/(x-y))]-2 L5[(1/(x-y))]-(5/3) L5[((-1+x)/(x-y))]+L5[((-1+x)/(x (x-y)))]+(11/3) L5[(x/(x-y))]+L5[((x^2)/(x-y))]+(5/9) L5[((x^2)/((-1+x) (x-y)))]-(1/3) L5[-((x-y)/((-1+x) x))]+L5[((x-y)/((-1+x) x))]+(5/9) L5[((x (x-y))/(((-1+x))^2))]-(1/6) L5[((((x-y))^2)/((((-1+x))^2) (x^2)))]+(1/9) L5[((((x-y))^2)/((-1+x) x (((-1+y))^3)))]+(5/9) L5[-((-1+x)/(((-1+y))^2))]+(1/6) L5[((((-1+x))^2)/((x^2) (((-1+y))^2)))]+L5[((x-y)/(((-1+y))^2))]-(5/9) L5[((x-y)/(x (((-1+y))^2)))]+(5/3) L5[-(((-1+x) (x-y))/(x (((-1+y))^2)))]+(1/48) L5[((((x-y))^2)/((((-1+x))^2) (((-1+y))^2)))]+(5/18) L5[((((x-y))^2)/(x (((-1+y))^2)))]-(11/3) L5[-(1/(-1+y))]-(4/9) L5[(1/((-1+x) (-1+y)))]-L5[-(1/(x (-1+y)))]+L5[-((-1+x)/(x (-1+y)))]+L5[((((-1+x))^2)/(x (-1+y)))]-(1/9) L5[(x/((((x-y))^2) (-1+y)))]-(1/9) L5[((((-1+x))^3)/((x^2) (x-y) (-1+y)))]+(4/9) L5[-(x/((x-y) (-1+y)))]+(7/3) L5[-((x-y)/((-1+x) (-1+y)))]-2 L5[((x-y)/((-1+x) (-1+y)))]+(4/9) L5[-((x-y)/((x^2) (-1+y)))]+(4/3) L5[-((x-y)/(x (-1+y)))]-(5/9) L5[-((((x-y))^2)/((x^2) (-1+y)))]-L5[-((((x-y))^2)/(x (-1+y)))]-L5[-((((x-y))^2)/((-1+x) x (-1+y)))]+(1/9) L5[-((((x-y))^3)/((-1+x) (x^3) (-1+y)))]+(5/9) L5[-((-1+y)/(((-1+x))^2))]-(8/3) L5[((-1+y)/(-1+x))]+(1/3) L5[-((-1+y)/x)]-L5[((-1+y)/x)]-L5[((x (-1+y))/(-1+x))]+(4/3) L5[-((x (-1+y))/((-1+x) (x-y)))]-2 L5[((x (-1+y))/((-1+x) (x-y)))]+L5[(((-1+x) x (-1+y))/(x-y))]+(1/48) L5[((((-1+y))^2)/(x^2))]+(1/48) L5[(((x^2) (((-1+y))^2))/((((-1+x))^2) (((x-y))^2)))]-(1/9) L5[((x (((-1+y))^2))/((((-1+x))^3) (x-y)))]+(1/9) L5[-((x (x-y))/((((-1+x))^2) (y^3)))]-(1/48) L5[((x^2)/((((-1+x))^2) (y^2)))]-(1/9) L5[((((-1+x))^2)/((x-y) (y^2)))]-(5/9) L5[((x-y)/((-1+x) (y^2)))]-(5/9) L5[-((((x-y))^2)/((-1+x) (y^2)))]+(1/6) L5[((((x-y))^2)/((((-1+y))^2) (y^2)))]-(3/16) L5[(((x^2) (((-1+y))^2))/((((x-y))^2) (y^2)))]+(1/9) L5[-((x (((-1+y))^3))/((((-1+x))^3) (y^2)))]-(7/18) L5[(1/y)]+(103/54) L5[(1/(x y))]-(4/3) L5[-(x/((-1+x) y))]+(4/9) L5[((-1+x)/((x-y) y))]+(4/9) L5[-((x-y)/((((-1+x))^2) y))]+(4/3) L5[((x-y)/((-1+x) y))]-L5[((((x-y))^2)/((-1+x) x y))]+(1/9) L5[-((((x-y))^3)/((((-1+x))^3) x y))]-(1/9) L5[(((-1+x) (x-y))/((((-1+y))^3) y))]-(5/27) L5[((((x-y))^3)/(x (((-1+y))^3) y))]+L5[((-1+x)/((-1+y) y))]+(1/9) L5[-((-1+x)/((x^2) (-1+y) y))]-L5[-((x-y)/((-1+y) y))]+2 L5[((x-y)/((-1+y) y))]-(1/9) L5[((((x-y))^2)/((((-1+x))^3) (-1+y) y))]+(5/3) L5[-((((x-y))^2)/((-1+x) (-1+y) y))]+(1/9) L5[-(((-1+x) (x^2) (-1+y))/((((x-y))^3) y))]+(1/3) L5[-((x (-1+y))/((x-y) y))]-(7/3) L5[-(y/(-1+x))]+3 L5[(y/(-1+x))]+(157/54) L5[(y/(x^2))]-(1/6) L5[(y/x)]+L5[-(y/((-1+x) x))]+L5[(((-1+x) y)/x)]-L5[((x y)/(-1+x))]-(1/9) L5[-(((x^3) y)/((-1+x) (((x-y))^2)))]-(8/3) L5[-(y/(x-y))]+2 L5[-(((-1+x) y)/(x (x-y)))]-L5[(((-1+x) y)/(x (x-y)))]+L5[(((((-1+x))^2) y)/(x (x-y)))]+(5/3) L5[-((x y)/(x-y))]-(2/3) L5[((x y)/(x-y))]+L5[-((x y)/((-1+x) (x-y)))]-(1/9) L5[-(((x-y) y)/((-1+x) (x^3)))]+L5[(((x-y) y)/x)]-L5[(((x-y) y)/((-1+x) x))]-(5/27) L5[-(((((-1+x))^3) y)/((x^2) (((-1+y))^3)))]-(1/9) L5[(((((x-y))^2) y)/((-1+x) (((-1+y))^3)))]+(1/9) L5[(((((x-y))^3) y)/((x^2) (((-1+y))^3)))]+(1/9) L5[(y/((-1+x) x (((-1+y))^2)))]+(7/18) L5[(((((-1+x))^2) y)/(x (((-1+y))^2)))]-(1/9) L5[(((x-y) y)/((x^3) (((-1+y))^2)))]-L5[-(((x-y) y)/(x (((-1+y))^2)))]-L5[-(((-1+x) (x-y) y)/(x (((-1+y))^2)))]+(11/3) L5[(y/(-1+y))]-L5[-(y/((-1+x) (-1+y)))]-4 L5[-(((-1+x) y)/(-1+y))]-(2/3) L5[(((-1+x) y)/(-1+y))]+L5[(((-1+x) y)/((x^2) (-1+y)))]+(5/9) L5[(((((-1+x))^2) y)/((x^2) (-1+y)))]+(7/3) L5[-(y/(x (-1+y)))]-3 L5[(y/(x (-1+y)))]-(2/3) L5[(((-1+x) y)/(x (-1+y)))]-(4/9) L5[((x y)/((-1+x) (-1+y)))]+L5[(((x-y) y)/((-1+x) (-1+y)))]-L5[-(((x-y) y)/((x^2) (-1+y)))]-(4/9) L5[(((-1+y) y)/((-1+x) x))]-L5[-(((-1+y) y)/((-1+x) (x-y)))]-L5[-(((-1+x) (-1+y) y)/(x-y))]+(1/9) L5[((x (((-1+y))^2) y)/((-1+x) (((x-y))^3)))]-(1/48) L5[((y^2)/(((-1+x))^2))]-(107/54) L5[((y^2)/x)]+(1/6) L5[(((((-1+x))^2) (y^2))/((x^2) (((x-y))^2)))]+(1/48) L5[((y^2)/((x^2) (((-1+y))^2)))]-(4/9) L5[(((-1+x) (y^2))/(x (((-1+y))^2)))]-(4/9) L5[(((-1+x) (y^2))/((x^2) (-1+y)))]+L5[((y^2)/(x (-1+y)))]-L5[(((-1+x) (y^2))/(x (-1+y)))]+(1/9) L5[-((x (y^2))/((((-1+x))^2) (-1+y)))]-(1/9) L5[(((((-1+x))^3) (y^2))/((x^3) (x-y) (-1+y)))]-L5[-(((-1+x) (y^2))/(x (x-y) (-1+y)))]+(1/9) L5[-(((-1+y) (y^2))/((((-1+x))^2) x))]+(1/9) L5[-(((-1+x) (x-y) (y^3))/((x^2) (((-1+y))^3)))]+(1/9) L5[(((-1+x) (y^3))/((x^3) (((-1+y))^2)))]-(1/9) L5[(((x-y) (y^3))/((x^2) (-1+y)))]-(1/9) L5[((-1+y)/(x (((-x+y))^2)))]-(5/9) L5[(((-1+y) y)/(((-x+y))^2))]-(2/3) L5[((-1+y)/(-x+y))]+L5[-((-1+y)/((-1+x) (-x+y)))]+(4/9) L5[((-1+y)/(x (-x+y)))]+(4/9) L5[-(y/((-1+x) (-x+y)))]+L5[-(y/((-1+y) (-x+y)))]-(5/3) L5[-(((-1+x) y)/((-1+y) (-x+y)))]+(2/3) L5[(((-1+x) y)/((-1+y) (-x+y)))]+(5/3) L5[-(((((-1+x))^2) y)/(x (-1+y) (-x+y)))]-(1/9) L5[-((y^2)/((((-1+x))^2) (-x+y)))]+(4/9) L5[((y^2)/((-1+y) (-x+y)))]-(1/9) L5[-((y^3)/(x (((-1+y))^2) (-x+y)))]-L5[-(((-1+y) (-x+y))/((((-1+x))^2) y))]+(4/9) L5[((y (-x+y))/(((-1+y))^2))]+(5/18) L5[((((-x+y))^2)/((((-1+x))^2) y))]+(1/9) L5[(((-1+y) (((-x+y))^3))/((((-1+x))^2) (y^3)))]-(5/27) L5[((x (((-x+y))^3))/((((-1+x))^3) (y^2)))]//CreateTensor//sh//flatTensor//Expand

-N5[-(1/(-1+x))]+N5[((-1+x)/(x-y))]+N5[-(1/(-1+y))]-N5[-((x-y)/(x (-1+y)))]+N5[-(y/(x-y))]//CreateTensor//sh//flatTensor//Expand

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

I32[cr[{a,b,c,d,e}]] + I32[cr[{a,b,d,c,e}]] + I41[cr[{a,b,c,d,e}]] + I41[cr[{a,b,d,e,c}]] + I41[cr[{a,b,e,c,d}]] - (2L5[cr[{b,a,d,e}]] - 4L5[cr[{a,b,c,d}]]  + 2L5[cr[{a,b,c,e}]])//CreateTensor//sh//flatTensor//Expand

I32[cr[{a,b,c,d,e}]] + I32[cr[{a,b,c,e,d}]] + 3 I41[cr[{a,b,c,d,e}]]//CreateTensor//sh//flatTensor//Expand

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

args2 = Permutations[{a,b,c,d,e}];

basgen[i_,n_] := Table[If[k==i,1,0],{k,1,n}];

lookupI32[l___]:=basgen[Position[args2,{l}][[1,1]],Length[args2]];

id2[a_,b_,c_,d_,e_] := I32[a,b,c,d,e] + I32[b,a,c,d,e];

id4[a_,b_,c_,d_,e_] := I32[a,b,c,d,e]+ I32[a,b,d,c,e] - I32[a,b,c,e,d] - I32[a,b,e,c,d];

id6[a_,b_,c_,d_,e_] := I32[a,b,c,d,e] + I32[b,c,a,d,e] + I32[c,a,b,d,e] + I32[a,b,c,e,d] + I32[b,c,a,e,d] + I32[c,a,b,e,d]

Sx[{a_,b_,c_,d_,e_}] := I32[a,b,c,d,e] + I32[b,c,a,d,e] + I32[c,a,b,d,e]

id30[a_,b_,c_,d_,e_] := (Sx[{a,b,c,d,e}] + Sx[{b,c,d,e,a}] + Sx[{c,d,e,a,b}] + Sx[{d,e,a,b,c}] + Sx[{e,a,b,c,d}]) - 
(Sx[{a,c,e,b,d}] + Sx[{c,e,b,d,a}] + Sx[{e,b,d,a,c}] + Sx[{b,d,a,c,e}] + Sx[{d,a,c,e,b}]);

id2tbl = id2@@@Permutations[{a,b,c,d,e}]/.I32->lookupI32//Union//RowReduce//DeleteCases[#,el_/;VectorQ[el,#==0&]]&;

id4tbl = id4@@@Permutations[{a,b,c,d,e}]/.I32->lookupI32//Union//RowReduce//DeleteCases[#,el_/;VectorQ[el,#==0&]]&;

id6tbl = id6@@@Permutations[{a,b,c,d,e}]/.I32->lookupI32//Union//RowReduce//DeleteCases[#,el_/;VectorQ[el,#==0&]]&;

id30tbl = id30@@@Permutations[{a,b,c,d,e}]/.I32->lookupI32//Union//RowReduce//DeleteCases[#,el_/;VectorQ[el,#==0&]]&;

Length/@{id2tbl,id4tbl,id6tbl, id30tbl}

MatrixRank[Join[id2tbl,id4tbl,id6tbl,id30tbl]]

id2[Inf,0,1,x,y]/.{I32[l___]:>I32[cr[{l}]]}//CreateTensor//del//Expand

id4[Inf,0,1,x,y]/.{I32[l___]:>I32[cr[{l}]]}//CreateTensor//del//Expand

id6[Inf,0,1,x,y]/.{I32[l___]:>I32[cr[{l}]]}//CreateTensor//del//Expand

id30[Inf,0,1,x,y]/.{I32[l___]:>I32[cr[{l}]]}//CreateTensor//del//Expand