Rules for integrands of the form(((c+d x))^m) (((a+b Tanh[e+f x]))^n)

N:  ∫(u^m)(((a+b Tanh[v]))^n)dx  when  u==c+d x ∧ v==e+f x

Derivation: Algebraic normalization

Rule: If  u==c+d x ∧ v==e+f x, then

∫(u^m)(((a+b Tanh[v]))^n)dx  ⟶  ∫(((c+d x))^m) (((a+b Tanh[e+f x]))^n)dx

Program code:

Int[u_^m_.*(a_.+b_.*Tanh[v_])^n_.,x_Symbol] :=
  Int[ExpandToSum[u,x]^m*(a+b*Tanh[ExpandToSum[v,x]])^n,x] /;
FreeQ[{a,b,m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]]

Int[u_^m_.*(a_.+b_.*Coth[v_])^n_.,x_Symbol] :=
  Int[ExpandToSum[u,x]^m*(a+b*Coth[ExpandToSum[v,x]])^n,x] /;
FreeQ[{a,b,m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]]