MyCollect[ℰ_,vs_List]:=MyCollect[ℰ,vs,Identity];
MyCollect[ℰ_,vs_List,simp_]:=Total[CoefficientRules[ℰ,vs]/.((ps_->c_):>simp[c]Times@@((vs^ps)))];
MyCollect[ℰs_List,vs_List] := MyCollect[#,vs]&/@ℰs;
MyCollect[ℰs_List,vs_List,simp_] := MyCollect[#,vs,simp]&/@ℰs;
MyCollect[sd_SeriesData,vs_List] := MapAt[MyCollect[#,vs]&, sd, 3];
MyCollect[sd_SeriesData,vs_List,simp_] := MapAt[MyCollect[#,vs,simp]&, sd, 3];

r=(ω^(-1))+(((-2T ω dω)/((T-1)))x y+2T ω dω a +(ω T dω-((((T-1))^2)/T)p1))(ϵ/(ω^3))+(2 T (ω^2)(2 ((dω)^2) T-dω ω-ddω T ω)(a^2)+(((2 (-1+T) p1 ((1+T) ω-3 (-1+T) T dω))/T)+2 ω ((((-1+T))^2)dp1+2 (T^2) (((dω))^2)-T ω(dω+T ddω)))ω a+((T (ω^2)(4 ((dω)^2) (-1+T) T-dω (-3+T) ω-2 ddω (-1+T) T ω))/((((-1+T))^3) ))(x^2)(y^2)-((2 ω(-3 dω p1 (-1+T) T+dp1 (-1+T) T ω+p1 (1+T) ω))/(T ))x y+((4 T (ω^2)(2 ((dω)^2) (1-T) T-dω ω-ddω (1-T) T ω))/((((-1+T))^2) ))a x y-((p2/2)+ω (3 dω p1 (((-1+T))^2)-(((p1 (-1+(T^2))+T (dp1 (((-1+T))^2)+2 ((dω)^2) (T^2))) ω)/T)+T (dω+ddω T) (ω^2))))((ϵ^2)/(ω^5))

Res1=FullSimplify@Log[ω r+((O[ϵ])^3)]

TeXForm[Res1 /. {dω->ω',ddω->ω'',p1->(Subscript[ρ, 1]),dp1->(Subscript[ρ, 1])', p2->(Subscript[ρ, 2])}]

FullSimplify[Res1/.T->1+T]

Res2=MyCollect[Log[ω r+((O[ϵ])^3)], {a,x,y},FullSimplify]

MyCollect[Log[(ω r/.ϵ->(ω^2)ϵ)+((O[ϵ])^3)], {ω,dω,ddω,p1,dp1,p2},Simplify]

MyCollect[Log[(ω r/.ϵ->(ω^2)ϵ)+((O[ϵ])^3)], {a,x,y},FullSimplify]

MyCollect[(ω^5) r+((O[ϵ])^3), {ω,dω,ddω,p1,dp1,p2},Simplify]

MyCollect[Log[(ω r/.ϵ->(ω^2)ϵ)+((O[ϵ])^3)] /. {p1->((T ρ1)/(((T-1))^2)), dp1->((-(1+T) ρ1+(-1+T) T dρ1)/(((-1+T))^3))}, {a,x,y},FullSimplify]

FullSimplify[(∂_T)(((T ρ1[T])/(((T-1))^2)))]