(*******************************************************************) (* This file contains the {5,1} coproducts of V^{(3)}(u,v,w) *) (* and \tilde{V}^{(3)}(u,v,w) in terms of the weight 5 hexagon *) (* functions defined in arXiv:1308.2276. *) (* We only give the independent ones, given the *) (* (anti) symmetry under u <-> w and final entry conditions. *) (* *) (* We use a shorthand for the harmonic polylogarithms, so that *) (* H41m(u) = HPL[{4,1},1-u], etc. *) (* Also, tPhi6 = \tilde\Phi_6, F1 = F_1, Omega2t = \Omega6{(2)}, *) (* M1t = M_1 and Qu = Q_{ep}, *) (*******************************************************************) (* independent {5,1} coproducts of V^{(3)}(u,v,w) *) (* [ V^{(3)} ]^u *) V3[u] := 1/32 * ( - M1t(u,v,w) + M1t(v,u,w) + 32/3 * ( - 5 * ( Qu(v,w,u) - Qu(v,u,w) ) + Qu(w,u,v) - Qu(w,v,u) ) + ( 5 * H1m(u) + 3 * H1m(v) - 4 * H1m(w) ) * Omega2t(u,v,w) + ( 3 * H1m(u) + 5 * H1m(v) - 4 * H1m(w) ) * Omega2t(v,w,u) + ( 7 * H1m(u) + 5 * H1m(v) - 4 * H1m(w) ) * Omega2t(w,u,v) - 24 * ( H5m(u) - H41m(u) ) + 5 * H32m(u) + 12 * H311m(u) + 9 * H221m(u) + H2m(u) * ( 8 * H3m(u) - 11 * H21m(u) ) + H1m(u) * ( 8 * ( H4m(u) + H31m(u) ) - 15 * H211m(u) - 3 * H2m(u)^2 + 15 * H1m(u) * H3m(u) + 5/2 * H1m(u)^2 * H2m(u) ) + 2 * H41m(v) + 5/2 * H32m(v) - 126 * H311m(v) - 83/2 * H221m(v) - 4 * H2111m(v) + H2m(v) * ( - H3m(v) + 49/2 * H21m(v) ) + H1m(v) * ( 18 * H4m(v) + 13 * H31m(v) + 1/2 * H211m(v) - 53/4 * H2m(v)^2 + H1m(v) * ( 33/2 * H3m(v) - 6 * H21m(v) + 37/12 * H1m(v) * H2m(v) ) ) + 24 * H5m(w) - 26 * H41m(w) - 15/2 * H32m(w) + 210 * H311m(w) + 129/2 * H221m(w) + 20 * H2111m(w) + H2m(w) * ( H3m(w) - 83/2 * H21m(w) ) + H1m(w) * ( - 6 * H4m(w) - 33 * H31m(w) - 11/2 * H211m(w) + 73/4 * H2m(w)^2 + H1m(w) * ( - 31/2 * H3m(w) + 6 * H21m(w) - 35/12 * H1m(w) * H2m(w) ) ) + 1/6 * H2m(u) * ( 5 * H3m(v) + 17 * H21m(v) ) + 1/6 * H2m(v) * ( 67 * H3m(u) - 41 * H21m(u) ) + H1m(u) * ( 18 * H4m(v) + 16 * H31m(v) + 2 * H211m(v) - 33/2 * H2m(v)^2 - 53/6 * H2m(u) * H2m(v) + H1m(v) * ( 14 * H3m(v) - 8 * H21m(v) + 20 * H3m(u) - 2 * H21m(u) + H1m(v) * ( H2m(v) - 11/12 * H2m(u) ) ) + H1m(u) * ( - 19/12 * H3m(v) - 103/12 * H21m(v) + H1m(v) * ( 2 * H2m(u) - 25/12 * H2m(v) - 29/24 * H1m(v)^2 ) + H1m(u) * ( - 5/4 * H2m(v) - 9/8 * H1m(v)^2 ) ) ) + H1m(v) * ( 8 * H4m(u) + 42 * H31m(u) - 8 * H211m(u) - 37/2 * H2m(u)^2 - 67/6 * H2m(u) * H2m(v) + H1m(v) * ( 19/12 * H3m(u) - 17/12 * H21m(u) + 11/12 * H1m(v) * H2m(u) ) ) + 1/6 * H2m(u) * ( H3m(w) - 59 * H21m(w) ) + 1/6 * H2m(w) * ( 47 * H3m(u) - 13 * H21m(u) ) + H1m(u) * ( 4 * H4m(w) - 30 * H31m(w) - 28 * H211m(w) + 9/2 * H2m(w)^2 - 25/6 * H2m(u) * H2m(w) + H1m(w) * ( - 20 * H3m(u) + 2 * H21m(u) + 4 * H3m(w) + 12 * H21m(w) + H1m(w) * ( 5/12 * H2m(u) + H2m(w) ) ) + H1m(u) * ( - 23/12 * H3m(w) + 133/12 * H21m(w) + H1m(w) * ( - 2 * H2m(u) - 11/12 * H2m(w) + 17/24 * H1m(w)^2 ) + H1m(u) * ( - 5/4 * H2m(w) - 9/8 * H1m(w)^2 ) ) ) + H1m(w) * ( - 8 * H4m(u) - 40 * H31m(u) + 8 * H211m(u) + 18 * H2m(u)^2 + 85/6 * H2m(u) *H2m(w) + H1m(w) * ( - 1/12 * H3m(u) + 35/12 * H21m(u) - 1/4 * H1m(w) * H2m(u) ) ) - 4 * H2m(v) * ( H3m(w) + 3 * H21m(w) ) + 4 * H2m(w) * H21m(v) + H1m(v) * ( 32 * H4m(w) + 26 * H31m(w) + 16 * H211m(w) - 45/2 * H2m(w)^2 - 16 * H2m(v) * H2m(w) + H1m(w) * ( - 14 * H3m(v) + 8 * H21m(v) + 28 * H3m(w) - 20 * H21m(w) + 9/2 * H1m(w) * ( H2m(v) + 2 * H2m(w) ) ) + H1m(v) * ( 18 * H21m(w) - H1m(w) * ( 7 * H2m(v) + 11 * H2m(w) ) + 4/3 * H1m(w)^3 ) ) - H1m(v)^3 * ( 7/6 * H2m(w) + 9/4 * H1m(w)^2 ) + H1m(w) * ( - 18 * ( H4m(v) + H31m(v) ) - 2 * H211m(v) + 13 * H2m(v)^2 + 14 * H2m(v) * H2m(w) - 2 * H1m(w) * ( H3m(v) + 5 * H21m(v) ) ) + H1m(w)^3 * H2m(v) + H1m(u) * ( - 12 * H2m(v) * H2m(w) - 8/3 * H1m(w) * H1m(v)^3 + H1m(v)^2 * ( - 21/2 * H2m(w) - 23/4 * H1m(w)^2 ) + H1m(v) * ( - 40 * H21m(w) + 7/3 * H1m(w)^3 + H1m(w) * ( - 6 * H2m(u) - 20 * H2m(v) + 10 * H2m(w) ) ) + H1m(w) * ( 4 * H3m(v) + 22 * H21m(v) ) + 11/2 * H1m(w)^2 * H2m(v) ) + H1m(u)^2 * ( - 5/2 * H1m(v)^2 * H1m(w) + H1m(w) * H2m(v) + H1m(v) * ( - 3/2 * H2m(w) + 9/4 * H1m(w)^2 ) ) - 12 * ( H1m(v)^2 - H2m(v) ) * H1m(w) * H2m(u) + H1m(v) * ( - 16 * H2m(u) * H2m(w) + 19/2 * H1m(w)^2 * H2m(u) + H1m(w) * ( 8 * H3m(u) - 6 * H21m(u) + H1m(u)^3 ) ) + Zeta[2] * ( - 51/2 * H3m(u) + 1/4 * H3m(v) + 37/4 * H3m(w) + 10 * H21m(u) - 47 * H21m(v) + 93 * H21m(w) + ( 17/2 * H1m(u) + 70 * H1m(v) - 68 * H1m(w) ) * H2m(u) + ( 46 * H1m(u) + 201/4 * H1m(v) - 56 * H1m(w) ) * H2m(v) + ( - 22 * H1m(u) + 78 * H1m(v) - 267/4 * H1m(w) ) * H2m(w) - 7/2 * H1m(u)^3 - 77/12 * H1m(v)^3 + 55/12 * H1m(w)^3 + ( 14 * H1m(v)^2 - 18 * H1m(w)^2 ) * H1m(u) + ( - 34 * H1m(w)^2 - 4 * H1m(u)^2 ) * H1m(v) + ( 4 * H1m(u)^2 + 46 * H1m(v)^2 ) * H1m(w) + 16 * H1m(u) * H1m(v) * H1m(w) ) + Zeta[3] * ( - 22 * H2m(u) + 7 * H2m(v) + 15 * H2m(w) - 9 * H1m(u)^2 + 3/2 * H1m(v)^2 + 15/2 * H1m(w)^2 ) + Zeta[4] * ( - 10 * H1m(u) - 462 * H1m(v) + 468 * H1m(w) ) ) : (* [ V^{(3)} ]^{1-u} = - [ V^{(3)} ]^u *) (* - [ V^{(3)} ]^v - [ V^{(3)} ]^{1-v} *) V3[1-u] := - V3[u] - V3[v] - V3[1-v] : (* [ V^{(3)} ]^v is symmetric, so it can be written as *) (* V3vA(u,v,w) + V3vA(w,v,u), where V3vA(u,v,w) is given by: *) V3vA := 1/32 * ( M1t(u,v,w) - M1t(v,w,u) + 32/3 * ( Qu(u,v,w) - Qu(w,u,v) ) - ( 3 * ln(u) + 4 * ln(v) + 5 * ln(w) ) * Omega2t(u,v,w) - ( 5 * ln(u) + 3 * ln(v) ) * Omega2t(w,u,v) - 24 * H5m(u) + 18 * H41m(u) + 5/2 * H32m(u) - 294 * H311m(u) - 179/2 * H221m(u) + 28 * H2111m(u) + H2m(u) * ( 3 * H3m(u) + 97/2 * H21m(u) ) + H1m(u) * ( - 2 * H4m(u) + 57 * H31m(u) - 55/2 * H211m(u) - 69/4 * H2m(u)^2 + H1m(u) * ( 17/2 * H3m(u) + 6 * H21m(u) ) + 5/12 * H1m(u)^2 * H2m(u) ) - 24 * H5m(v) + 14 * H41m(v) - 5/2 * H32m(v) - 42 * H311m(v) - 13/2 * H221m(v) + 36 * H2111m(v) + 9/2 * H2m(v) * ( 2 * H3m(v) - H21m(v) ) + H1m(v) * ( - 14 * H4m(v) + 19 * H31m(v) - 57/2 * H211m(v) + 9/4 * H2m(v)^2 - 1/2 * H1m(v) * ( H3m(v) - 12 * H21m(v) ) + 1/4 * H1m(v)^2 * H2m(v) ) + 1/6 * H2m(u) * ( 91 * H3m(v) - 89 * H21m(v) ) + 1/6 * H2m(v) * ( 29 * H3m(u) + 41 * H21m(u) ) + H1m(u) * ( 16 * H4m(v) - 6 * H31m(v) - 13/2 * H2m(v)^2 - 55/6 * H2m(u) * H2m(v) + H1m(v) * ( 12 * H3m(u) + 2 * H21m(u) + 24 * H3m(v) - 16 * H21m(v) + H1m(v) * ( 47/12 * H2m(u) + 8 * H2m(v) ) ) ) + H1m(u)^2 * ( 43/12 * H3m(v) - 65/12 * H21m(v) - 7/8 * H1m(v)^3 + H1m(v) * ( H2m(u) - 5/12 * H2m(v) ) ) - H1m(u)^3 * ( 7/4 * H2m(v) + 29/24 * H1m(v)^2 ) + H1m(v) * ( 16 * H4m(u) - 36 * H31m(u) - 16 * H211m(u) + 3 * H2m(u)^2 + 19/6 * H2m(u) * H2m(v) ) + 1/12 * H1m(v)^2 * ( 29 * H3m(u) - 199 * H21m(u) ) - 7/4 * H1m(v)^3 * H2m(u) + 4 * H2m(u) * H3m(w) + H1m(u) * ( 20 * H4m(w) + 30 * H31m(w) + 4 * H211m(w) - 41/2 * H2m(w)^2 - 10 * H2m(u) * H2m(w) + 10 * H1m(w) * ( 2 * H3m(w) - H21m(w) ) + H1m(w)^2 * ( 21/2 * H2m(u) + 5 * H2m(w) ) ) - H1m(u)^2 * ( 2 * H3m(w) + 22 * H21m(w) + 7/12 * H1m(w)^3 ) - 1/2 * H1m(u)^3 * H2m(w) - 4 * H2m(u) * H2m(w) * H1m(v) + H1m(u) * ( - 12 * H2m(v) * H2m(w) + H1m(v) * ( 4 * H3m(w) + 42 * H21m(w) + 2/3 * H1m(w)^3 - H1m(w) * ( 10 * H2m(v) + 32 * H2m(u) ) ) - H1m(v)^2 * ( 3/2 * H2m(w) + 35/4 * H1m(w)^2 ) + H1m(w) * ( 6 * H21m(v) - 4 * H3m(v) ) - 1/2 * H1m(w)^2 * H2m(v) ) - H1m(u)^2 * H1m(v) * ( 10 * H2m(w) - 5/4 * H1m(w)^2 ) + Zeta[2] * ( - 31/4 * H3m(u) - 65/4 * H3m(v) - 95 * H21m(u) + 7 * H21m(v) + ( 265/4 * H1m(u) - 28 * H1m(v) + 66 * H1m(w) ) * H2m(u) + ( 26 * H1m(u) - 25/4 * H1m(v) ) * H2m(v) - 13/12 * H1m(u)^3 - 1/4 * H1m(v)^3 - 16 * H1m(v)^2 * H1m(u) + 14 * H1m(u)^2 * H1m(v) - 26 * H1m(u)^2 * H1m(w) + 60 * H1m(u) * H1m(v) * H1m(w) ) + Zeta[3] * ( 7 * H2m(u) - 7 * H2m(v) + 3/2 * H1m(u)^2 - 3/2 * H1m(v)^2 ) + Zeta[4] * ( - 378 * H1m(u) + 118 * H1m(v) ) ) : (* Add the image under u <-> w *) V3[v] := V3vA + flip_uw(V3vA) : (* V3^{1-v}(u,v,w) is also symmetric; we write it as *) V3[1-v] := - V3[v] - V3omvA - flip_uw(V3omvA) : (* where V3omvA is given by *) V3omvA := 1/32 * ( - 4 * ( H1m(u) - H1m(v) + H1m(w) ) * Omega2t(u,v,w) - 2 * ( 2 * H1m(u) - H1m(v) ) * Omega2t(w,u,v) + 24 * H5m(u) - 16 * H41m(u) + 264 * H311m(u) + 80 * H221m(u) - 16 * H2111m(u) - 4 * H2m(u) * ( H3m(u) + 11 * H21m(u) ) + H1m(u) * ( 4 * H4m(u) - 52 * H31m(u) + 20 * H211m(u) + 16 * H2m(u)^2 ) - 4 * H1m(u)^2 * ( 2 * H3m(u) + H21m(u) ) - 2/3 * H1m(u)^3 * H2m(u) + H1m(v) * ( 8 * H4m(v) - 4 * H31m(v) - 3 * H2m(v)^2 ) + 4 * H1m(v)^2 * ( 2 * H3m(v) - H21m(v) ) + 2 * H1m(v)^3 * H2m(v) - 4 * H2m(v) * ( H3m(u) + H21m(u) ) + H1m(u) * ( - 16 * H4m(v) + 8 * H31m(v) + 6 * H2m(v)^2 + 8 * H2m(u) * H2m(v) + 8 * H1m(v) * ( 2 * ( H3m(u) - H3m(v) ) - H21m(u) + H21m(v) ) - H1m(v)^2 * ( 12 * H2m(u) + 8 * H2m(v) ) ) + H1m(u)^2 * ( 4 * H1m(v) * H2m(u) - 2 * H1m(v)^3 ) + H1m(u)^3 * ( 2 * H2m(v) + 4/3 * H1m(v)^2 ) + H1m(v) * ( 20 * H4m(u) + 36 * H31m(u) + 4 * H211m(u) - 20 * H2m(u)^2 - 12 * H2m(u) * H2m(v) ) + 20 * H1m(v)^2 * H21m(u) - 4 * H2m(u) * ( H3m(w) + H21m(w) ) + H1m(u) * ( - 20 * H4m(w) - 36 * H31m(w) - 4 * H211m(w) + 20 * H2m(w)^2 + 12 * H2m(u) * H2m(w) - 8 * H1m(w) * ( 2 * H3m(u) - H21m(u) ) - 2 * H1m(w)^2 * ( 5 * H2m(u) + 2 * H2m(w) ) ) + H1m(u)^2 * ( 20 * H21m(w) + 1/3 * H1m(w)^3 ) + H1m(u) * ( 12 * H2m(v) * H2m(w) + H1m(v) * ( - 4 * H3m(w) - 44 * H21m(w) - 2/3 * H1m(w)^3 + H1m(w) * ( 32 * H2m(u) + 6 * H2m(v) ) ) - 6 * H1m(v)^2 * ( 2 * H2m(w) - H1m(w)^2 ) - 2 * H1m(v)^3 * H1m(w) ) + H1m(u)^2 * H1m(v) * ( 10 * H2m(w) - 1/2 * H1m(w)^2 ) - 8 * H2m(u) * H2m(w) * H1m(v) + Zeta[2] * ( 8 * H3m(u) + 88 * H21m(u) + ( - 64 * H1m(u) + 72 * H1m(v) - 72 * H1m(w) ) * H2m(u) + ( - 24 * H1m(u) + 12 * H1m(v) ) * H2m(v) + 4/3 * H1m(u)^3 - 4 * H1m(v)^3 + 48 * H1m(v)^2 * H1m(u) - 24 * H1m(u)^2 * H1m(v) + 24 * H1m(u)^2 * H1m(w) - 60 * H1m(u) * H1m(v) * H1m(w) ) + Zeta[4] * ( 384 * H1m(u) - 192 * H1m(v) ) ) : (* [ V^{(3)} ]^{y_u} *) V3[yu] := 1/32 * ( 7 * H_1(u,v,w) + 5 * H_1(v,w,u) + 3 * H_1(w,u,v) - 17/4 * J_1(u,v,w) - 7/4 * ( J_1(v,w,u) + J_1(w,u,v) ) + ( 3 * ( ln(u)^2 + ln(v)^2 + ln(w)^2 ) + 4 * ( ln(u) * ( ln(v) - ln(w) ) + ln(v) * ln(w) ) + 10 * ( H2m(u) + H2m(v) + H2m(w) ) - 70 * Zeta[2] ) * tPhi6(u,v,w) ) : (* [ V^{(3)} ]^{y_v} *) V3[yv] := 1/32 * ( 6 * H_1(u,v,w) + 4 * ( H_1(v,w,u) + H_1(w,u,v) ) - 3/2 * ( J_1(u,v,w) + J_1(v,w,u) + J_1(w,u,v) ) + 2 * ( ln(u)^2 + ln(v)^2 + ln(w)^2 + 2 * ( ln(v) * ( ln(u) + ln(w) ) - ln(u) * ln(w) ) + 4 * ( H2m(u) + H2m(v) + H2m(w) ) - 22 * Zeta[2] ) * tPhi6(u,v,w) ) : (* The remaining entries follow from the u <-> w "flip" symmetry *) V3[w] := flip_uw(V3[u]) : V3[1-w] := flip_uw(V3[1-u]) : V3[yw] := flip_uw(V3[yu]) : (******************************************************************) (* independent {5,1} coproducts of \tilde{V}^{(3)}(u,v,w) *) (* [ \tilde{V}^{(3)} ]^u *) Vt3[u] := 1/96 * ( - H_1(u,v,w) + H_1(v,w,u) + 3 * H_1(w,u,v) - 23/4 * J_1(u,v,w) - 13/4 * J_1(v,w,u) - 3/4 * J_1(w,u,v) - 6 * ln(u) * ( F1(u,v,w) - F1(w,u,v) ) + 3 * tPhi6(u,v,w) * ( 3 * ln(u)^2 + ln(v)^2 + ln(w)^2 + 2 * ( H2m(u) + H2m(v) + H2m(w) - ln(u) * ln(w) ) ) - 78 * Zeta[2] * tPhi6(u,v,w) ) : (* [ \tilde{V}^{(3)} ]^v *) Vt3[v] := 1/96 * ( 2 * ( H_1(v,w,u) - H_1(w,u,v) ) + 5/2 * ( J_1(v,w,u) - J_1(w,u,v) ) - 6 * ln(v) * ( F1(u,v,w) - F1(w,u,v) - ( ln(u) - ln(w) ) * tPhi6(u,v,w) ) ) : (* [ \tilde{V}^{(3)} ]^{1-u} = - [ \tilde{V}^{(3)} ]^u *) Vt3[1-u] := - Vt3[u] : (* [ \tilde{V}^{(3)} ]^{1-v} = - [ \tilde{V}^{(3)} ]^v *) Vt3[1-v] := - Vt3[v] : (* [ \tilde{V}^{(3)} ]^{y_u} *) Vt3[yu] := 1/96 * ( M1t(w,v,u) - M1t(v,w,u) + 3 * ( M1t(w,u,v) - M1t(u,w,v) ) - 64/3 * ( 2 * ( Qu(w,v,u) - Qu(w,u,v) ) + 7 * ( Qu(v,w,u) - Qu(v,u,w) ) ) + ( 3 * ln(u) + ln(v) - 4 * ln(w) ) * Omega2t(u,v,w) - ( 3 * ln(u) - ln(v) - 2 * ln(w) ) * ( Omega2t(w,u,v) - Omega2t(v,w,u) ) - 72 * H5m(u) + 72 * H41m(u) + 15 * H32m(u) + 36 * H311m(u) + 27 * H221m(u) - 9 * H2m(u) * H21m(u) + 3 * ln(u) * ( 12 * H4m(u) - 4 * H31m(u) + 3 * H211m(u) - H2m(u)^2 ) - 3 * ln(u)^2 * H3m(u) - 3/2 * ln(u)^3 * H2m(u) - 12 * H41m(v) - 10 * H32m(v) + 168 * H311m(v) + 58 * H221m(v) + 56 * H2111m(v) + 6 * H2m(v) * ( H3m(v) - 7 * H21m(v) ) + 1/2 * ln(v) * ( 24 * H4m(v) + 36 * H31m(v) + 68 * H211m(v) - 31 * H2m(v)^2 ) - 8 * ln(v)^2 * ( H3m(v) - H21m(v) ) + 2/3 * ln(v)^3 * H2m(v) + 72 * H5m(w) - 60 * H41m(w) - 5 * H32m(w) + 84 * H311m(w) + 11 * H221m(w) - 8 * H2111m(w) - 3 * H2m(w) * ( 2 * H3m(w) + 3 * H21m(w) ) - 1/2 * ln(w) * ( 96 * H4m(w) - 84 * H31m(w) + 38 * H211m(w) - H2m(w)^2 ) + ln(w)^2 * ( 11 * H3m(w) - 8 * H21m(w) ) - 7/6 * ln(w)^3 * H2m(w) - 1/4 * ln(u)^3 * ( 5 * ln(v)^2 - 2 * H2m(v) ) + 1/12 * ln(u)^2 * ( 3 * ln(v)^3 + 2 * ln(v) * ( 35 * H2m(v) + 24 * H2m(u) ) + 44 * H3m(v) + 164 * H21m(v) ) + ln(u) * ( ln(v)^2 * ( 4 * H2m(v) + 1/6 * H2m(u) ) + ln(v) * ( - 18 * (H3m(u) + H3m(v)) + 16 * H21m(u) + 12 * H21m(v) ) + 18 * H4m(v) + 32 * H31m(v) + 10 * H211m(v) - 37/2 * H2m(v)^2 - 20/3 * H2m(v) * H2m(u) ) + 1/6 * ln(v)^3 * H2m(u) + 1/3 * ln(v)^2 * ( 13 * H3m(u) + 7 * H21m(u) ) + ln(v) * ( 14 * H4m(u) + 16 * H31m(u) + 22 * H211m(u) - 31/2 * H2m(u)^2 - 28/3 * H2m(v) * H2m(u) ) + 2/3 * H2m(u) * ( 5 * H3m(v) - 7 * H21m(v) ) + 26/3 * H2m(v) * ( H3m(u) + H21m(u) ) - ln(u)^3 * ( ln(w)^2 - H2m(w) ) + 1/12 * ln(u)^2 * ( 15 * ln(w)^3 + 2 * ln(w) * ( 19 * H2m(w) - 24 * H2m(u) ) - 26 * H3m(w) + 70 * H21m(w) ) + 1/3 * ln(u) * ( 4 * ln(w)^2 * ( H2m(u) + 6 * H2m(w) ) + 6 * ln(w) * ( 9 * H3m(u) - 8 * H21m(u) - 12 * ( H3m(w)- H21m(w) ) ) + 72 * H4m(w) - 6 * H31m(w) + 96 * H211m(w) - 69/2 * H2m(w)^2 - 7 * H2m(u) * H2m(w) ) - 13/6 * ln(w)^3 * H2m(u) + 1/6 * ln(w)^2 * ( H3m(u) + 13 * H21m(u) ) - 1/3 * ln(w) * ( 42 * H4m(u) + 66 * ( H31m(u) + H211m(u) ) - 51 * H2m(u)^2 - H2m(u) * H2m(w) ) - 1/3 * H2m(u) * ( H3m(w) + 49 * H21m(w) ) + 1/3 * H2m(w) * ( H3m(u) + 37 * H21m(u) ) + 1/12 * ln(v)^3 * ( 7 * ln(w)^2 + 10 * H2m(w) ) + 1/6 * ln(v)^2 * ( 13 * ln(w)^3 + 6 * ln(w) * ( 7 * H2m(w) - 4 * H2m(v) ) - 48 * H3m(w) + 60 * H21m(w) ) - 1/2 * ln(v) * ( ln(w)^2 * ( 4 * H2m(w) - 25 * H2m(v) ) + 12 * ln(w) * ( 2 * H21m(v) - 3 * H3m(v) - 4 * ( H3m(w) - H21m(w) ) ) + 48 * H4m(w) - 20 * H31m(w) + 64 * H211m(w) - 19 * H2m(w)^2 + 12 * H2m(v) * H2m(w) ) - 1/3 * ln(w)^3 * H2m(v) + 2 * ln(w)^2 * ( H3m(v) + 7 * H21m(v) ) + ln(w) * ( - 18 * H4m(v) - 34 * H31m(v) - 10 * H211m(v) + 19 * H2m(v)^2 + 12 * H2m(v) * H2m(w) ) - 24 * H2m(w) * H21m(v) - 12 * H2m(v) * H3m(w) + 3 * ln(u)^3 * ln(v) * ln(w) - 1/4 * ln(u)^2 * ( 4 * ln(v)^2 * ln(w) + ln(v) * ( 17 * ln(w)^2 + 18 * H2m(w) ) + 24 * ln(w) * H2m(v) ) + ln(u) * ( - 1/3 * ln(v)^3 * ln(w) - 1/4 * ln(v)^2 * ( 3 * ln(w)^2 - 10 * H2m(w) ) + ln(v) * ( 4/3 * ln(w)^3 - 2 * ln(w) * ( 8 * H2m(w) + 3 * H2m(u) + 13 * H2m(v) ) + 4 * H3m(w) - 24 * H21m(w) ) + 13/2 * ln(w)^2 * H2m(v) - 2 * ln(w) * ( 2 * H3m(v) + 21 * H21m(v) ) - 12 * H2m(v) * H2m(w) ) - 3/2 * ln(w) * ( 4 * ln(v)^2 * H2m(u) - ln(v) * ( 9 * H2m(u) * ln(w) + 4 * H21m(u) ) - 8 * H2m(u) * H2m(v) ) + Zeta[2] * ( - 57/2 * H3m(u) + H3m(v) + 55/2 * H3m(w) - 18 * H21m(u) + 92 * H21m(v) + 46 * H21m(w) + 1/2 * H2m(u) * ( 45 * ln(u) + 92 * ln(v) - 104 * ln(w) ) + H2m(v) * ( 58 * ln(u) + 53 * ln(v) - 60 * ln(w) ) + 1/2 * H2m(w) * ( 76 * ln(u) - 12 * ln(v) - 7 * ln(w) ) - 3/2 * ln(u)^3 - 1/3 * ln(v)^3 + 35/6 * ln(w)^3 - 8 * ( ln(u)^2 * ( ln(v) - ln(w) ) + ln(v)^2 * ( ln(u) - ln(w) ) ) - 4 * ln(w)^2 * ( 4 * ln(u) + 11 * ln(v) ) + 72 * ln(u) * ln(v) * ln(w) ) - Zeta[3] * ( 66 * H2m(u) - 8 * H2m(v) - 58 * H2m(w) + 27 * ln(u)^2 - 6 * ln(v)^2 - 21 * ln(w)^2 ) - 2 * Zeta[4] * ( 117 * ln(u) + 123 * ln(v) - 114 * ln(w) ) ) : (* [ \tilde{V}^{(3)} ]^{y_v}(u,v,w) is antisymmetric under u <-> w, *) (* so we write it as: *) Vt3[yv] := Vt3yvA - flip_uw(Vt3yvA) : (* where *) Vt3yvA := 1/96 * ( - 3 * ( M1t(u,v,w) + M1t(v,w,u) ) + 2 * M1t(w,u,v) + 320/3 * ( Qu(u,v,w) + Qu(w,u,v) ) - ( ln(u) - 6 * ln(v) + 5 * ln(w) ) * Omega2t(u,v,w) + ln(u) * Omega2t(w,u,v) - 72 * H5m(u) + 48 * H41m(u) - 5 * H32m(u) + 84 * H311m(u) + 47 * H221m(u) + 64 * H2111m(u) + 3 * H2m(u) * ( 4 * H3m(u) - 11 * H21m(u) ) + ln(u) * ( 60 * H4m(u) - 24 * H31m(u) + 53 * H211m(u) - 16 * H2m(u)^2 ) - ln(u)^2 * ( 19 * H3m(u) - 16 * H21m(u) ) + 11/6 * ln(u)^3 * H2m(u) - ln(u)^3 * ln(v)^2 + 1/6 * ln(u)^2 * ( 25 * H3m(v) + H21m(v) - 7 * ln(v) * H2m(v) ) + 1/3 * H2m(u) * ( 25 * H3m(v) - 11 * H21m(v) ) + 2 * ln(u) * ( 14 * H4m(v) + 19 * H31m(v) + 22 * H211m(v) - 65/4 * H2m(v)^2 - ln(v) * ( 18 * H3m(v) - 16 * H21m(v) ) + 4 * ln(v)^2 * H2m(v) ) - 1/4 * ln(v)^3 * ( ln(u)^2 + 2 * H2m(u) ) + 1/6 * ln(v)^2 * ( 35 * H3m(u) + 47 * H21m(u) + 16 * ln(u) * H2m(u) ) + 2 * ln(v) * ( - 3 * H4m(u) + 17 * H31m(u) - 11 * H211m(u) - 7/2 * H2m(u)^2 + 3 * ln(u) * ( H3m(u) - 2 * H21m(u) ) - 2 * ln(u)^2 * H2m(u) ) + 1/3 * H2m(v) * ( 7 * ln(u)^3 + 11 * H3m(u) + 35 * H21m(u) - ( 29 * ln(u) + 13 * ln(v) ) * H2m(u) ) - 1/12 * ln(u)^3 * ( 19 * ln(w)^2 - 14 * H2m(w) ) - 11/2 * ln(u)^2 * ln(w) * H2m(w) + 2 * ln(u) * ( - 19/4 * H2m(w)^2 + 3 * ln(w) * ( H3m(w) - 2 * H21m(w) ) + ln(w)^2 * H2m(w) ) + 2 * ln(w) * ( 3 * H4m(u) - 22 * H31m(u) + 11 * H211m(u) ) + 2 * ln(w)^2 * ( 5 * H3m(u) + 2 * H21m(u) ) + 6 * H2m(w) * ( 2 * H3m(u) - 4 * H21m(u) - 3 * ln(u) * H2m(u) ) + 1/4 * ln(v)^2 * ln(u) * ( 6 * H2m(w) - 13 * ln(w)^2 ) + 3/2 * H2m(v) * ln(w) * ( 8 * H2m(u) - 13 * ln(u)^2 ) + 1/3 * ln(v) * ( - 5 * ln(u)^3 * ln(w) - 12 * ln(u)^2 * H2m(w) + 6 * ln(u) * ( 4 * H3m(w) + 9 * H21m(w) + 5 * ln(w) * H2m(w) ) ) + Zeta[2] * ( - 53/2 * H3m(u) + 46 * H21m(u) + H2m(u) * ( 113/2 * ln(u) + 20 * ln(v) - 54 * ln(w) ) + 98 * ln(u) * H2m(v) - 37/6 * ln(u)^3 + ln(u)^2 * ( 8 * ln(v) + 52 * ln(w) ) - 16 * ln(u) * ln(v)^2 ) + Zeta[3] * ( - 50 * H2m(u) - 15 * ln(u)^2 ) - 474 * Zeta[4] * ln(u) ) : (* The remaining entries follow from the u <-> w "flip" anti-symmetry *) Vt3[w] := - flip_uw(Vt3[u]) : Vt3[1-w] := - flip_uw(Vt3[1-u]) : Vt3[yw] := - flip_uw(Vt3[yu]) :