velocity:
Subscript[v, 0]+0.5*(Subscript[v, 1]-Subscript[v, 0])*(Erf[Subscript[a, 1]*t-k]+Erf[k])-0.5*(Subscript[v, 2]-Subscript[v, 1])*(Erf[Subscript[a, 2]*(t-\[Tau])-k]+Erf[k])

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velocity integral:
(t-0.5 t Erf[k]+0.5 t Erf[k-1. t Subscript[a, 1]]+(-0.282095 E^(-1. k^2+2. k t Subscript[a, 1]-1. t^2 Subsuperscript[a, 1, 2])-0.5 k Erf[k-1. t Subscript[a, 1]])/Subscript[a, 1]) Subscript[v, 0]+(1. t Erf[k]-0.5 t Erf[k-1. t Subscript[a, 1]]-0.5 t Erf[k+(-1. t+\[Tau]) Subscript[a, 2]]+0.5 \[Tau] Erf[k+(-1. t+\[Tau]) Subscript[a, 2]]+(0.282095 E^(-1. k^2+2. k t Subscript[a, 1]-1. t^2 Subsuperscript[a, 1, 2])+0.5 k Erf[k-1. t Subscript[a, 1]])/Subscript[a, 1]+(0.282095 E^(-1. t^2 Subsuperscript[a, 2, 2]+2. t Subscript[a, 2] (k+\[Tau] Subscript[a, 2])-1. (k+\[Tau] Subscript[a, 2])^2)+0.5 k Erf[k+(-1. t+\[Tau]) Subscript[a, 2]])/Subscript[a, 2]) Subscript[v, 1]+(-0.5 t Erf[k]+(0.5 t-0.5 \[Tau]) Erf[k+(-1. t+\[Tau]) Subscript[a, 2]]+(-0.282095 E^(-1. t^2 Subsuperscript[a, 2, 2]+2. t Subscript[a, 2] (k+\[Tau] Subscript[a, 2])-1. (k+\[Tau] Subscript[a, 2])^2)-0.5 k Erf[k+(-1. t+\[Tau]) Subscript[a, 2]])/Subscript[a, 2]) Subscript[v, 2]

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normalization term:
(0. +(-0.282095 E^(0. -1. k^2)-0.5 k Erf[0. +k])/Subscript[a, 1]) Subscript[v, 0]+(0. +0.5 \[Tau] Erf[k+(0. +\[Tau]) Subscript[a, 2]]+(0.282095 E^(0. -1. k^2)+0.5 k Erf[0. +k])/Subscript[a, 1]+(0.282095 E^(0. -1. (k+\[Tau] Subscript[a, 2])^2)+0.5 k Erf[k+(0. +\[Tau]) Subscript[a, 2]])/Subscript[a, 2]) Subscript[v, 1]+(0. +(0. -0.5 \[Tau]) Erf[k+(0. +\[Tau]) Subscript[a, 2]]+(-0.282095 E^(0. -1. (k+\[Tau] Subscript[a, 2])^2)-0.5 k Erf[k+(0. +\[Tau]) Subscript[a, 2]])/Subscript[a, 2]) Subscript[v, 2]

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position = velocity integral - normalization term + initial position