c'[u] = 
        alfa[u] t[u] + beta[u] n[u]  + 
          gama[u] b[u];
t'[u] = kapa[u] n[u];
n'[u] = -kapa[u] t[u]  +  
          tau[u] b[u];
b'[u] = -tau[u] n[u];

X[u, v] = 
        c[u] + r[u] n[u] Cos[v] + r[u] Sin[v] b[u];

uu = D[X[u, v], u]
vv = D[X[u, v], v]

uuu = D[uu, u];
uv = D[uu, v];
vvv = D[vv, v];

UU = {Coefficient[uu, t[u], 1], Coefficient[uu, n[u], 1], 
          Coefficient[uu, b[u], 1]};
VV = {Coefficient[vv, t[u], 1], Coefficient[vv, n[u], 1], 
          Coefficient[vv, b[u], 1]};

EE = UU . UU;
FF = UU . VV;
GG = VV . VV;

W = Simplify[EE GG - FF^2];

UUU = {Coefficient[uuu, t[u], 1], Coefficient[uuu, n[u], 1], 
          Coefficient[uuu, b[u], 1]};
UV = {Coefficient[uv, t[u], 1], Coefficient[uv, n[u], 1], 
          Coefficient[uv, b[u], 1]};
VVV = {Coefficient[vvv, t[u], 1], Coefficient[vvv, n[u], 1], 
          Coefficient[vvv, b[u], 1]};

ee = Det[{UU, VV, UUU}];
ff = Det[{UU, VV, UV}];
gg = Det[{UU, VV, VVV}];

kk = TrigReduce[ee gg - ff^2 + W^2];

Simplify[Coefficient[kk, Cos[4  v], 1]]
Simplify[Coefficient[kk, Sin[4  v], 1]]

Simplify[Coefficient[kk, Cos[3  v], 1]]
Simplify[Coefficient[kk, Sin[3  v], 1]]

Simplify[Coefficient[kk, Cos[2  v], 1]]
Simplify[Coefficient[kk, Sin[2  v], 1]]

Simplify[Coefficient[kk, Cos[v], 1]]
Simplify[Coefficient[kk, Sin[v], 1]]