(* 1. Set up *)

field={Γ,k,((a[t])^(n_)),((b[t])^(n_)),h0[t],h1[t],h0'[t],h1'[t],h0''[t],h1''[t],Derivative[3][h0][t],Derivative[3][h1][t],Derivative[4][h0][t],Derivative[4][h1][t]};

rep1={a'[t]->a[t]H[t],a''[t]->a[t]((H[t])^2)+a[t]H'[t],Derivative[3][a][t]->a[t]((H[t])^3)+3a[t]H[t]H'[t]+a[t]H''[t],Derivative[4][a][t]->a[t] ((H[t])^4)+6 a[t] ((H[t])^2) H'[t]+3 a[t] ((H'[t])^2)+4 a[t] H[t] H''[t]+a[t] Derivative[3][H][t],b'[t]->b[t]G[t],b''[t]->b[t]((G[t])^2)+b[t]G'[t],Derivative[3][b][t]->b[t]((G[t])^3)+3b[t]G[t]G'[t]+b[t]G''[t],Derivative[4][b][t]->b[t] ((G[t])^4)+6 b[t] ((G[t])^2) G'[t]+3 b[t] ((G'[t])^2)+4 b[t] G[t] G''[t]+b[t] Derivative[3][G][t]};

rep3={Derivative[5][H][t]-> (((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^5)(-120((Subscript[H, i])/((τ[t])^6))-55440Γ ((Subscript[H, i])^7)((13824 (477087-97732 (Sqrt[21])))/(8575 ((τ[t])^12)))),Derivative[5][G][t]->((-7+(Sqrt[21]))/14)(((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^5)(-120((Subscript[H, i])/((τ[t])^6))-55440Γ ((Subscript[H, i])^7)((20736 (200843-34423 (Sqrt[21])))/(8575 ((τ[t])^12)))),Derivative[4][H][t]-> (((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^4)(24((Subscript[H, i])/((τ[t])^5))+5040Γ ((Subscript[H, i])^7)((13824 (477087-97732 (Sqrt[21])))/(8575 ((τ[t])^11)))),Derivative[4][G][t]->((-7+(Sqrt[21]))/14)(((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^4)(24((Subscript[H, i])/((τ[t])^5))+5040Γ ((Subscript[H, i])^7)((20736 (200843-34423 (Sqrt[21])))/(8575 ((τ[t])^11)))),Derivative[3][H][t]-> (((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^3)(-6((Subscript[H, i])/((τ[t])^4))-504Γ ((Subscript[H, i])^7)((13824 (477087-97732 (Sqrt[21])))/(8575 ((τ[t])^10)))),Derivative[3][G][t]->((-7+(Sqrt[21]))/14)(((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^3)(-6((Subscript[H, i])/((τ[t])^4))-504Γ ((Subscript[H, i])^7)((20736 (200843-34423 (Sqrt[21])))/(8575 ((τ[t])^10)))),H''[t]-> (((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^2)(2((Subscript[H, i])/((τ[t])^3))+56Γ ((Subscript[H, i])^7)((13824 (477087-97732 (Sqrt[21])))/(8575 ((τ[t])^9)))),G''[t]->((-7+(Sqrt[21]))/14)(((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^2)(2((Subscript[H, i])/((τ[t])^3))+56Γ ((Subscript[H, i])^7)((20736 (200843-34423 (Sqrt[21])))/(8575 ((τ[t])^9)))),H'[t]-> ((-1+(Sqrt[21]))/2)(Subscript[H, i])(-((Subscript[H, i])/((τ[t])^2))-7Γ ((Subscript[H, i])^7)((13824 (477087-97732 (Sqrt[21])))/(8575 ((τ[t])^8)))),G'[t]->((-7+(Sqrt[21]))/14)((-1+(Sqrt[21]))/2)(Subscript[H, i])(-((Subscript[H, i])/((τ[t])^2))-7Γ ((Subscript[H, i])^7)((20736 (200843-34423 (Sqrt[21])))/(8575 ((τ[t])^8)))),H[t]->((Subscript[H, i])/(τ[t]))+Γ ((Subscript[H, i])^7)((13824 (477087-97732 (Sqrt[21])))/(8575 ((τ[t])^7))),G[t]->((-7+(Sqrt[21]))/14)(((Subscript[H, i])/(τ[t]))+Γ ((Subscript[H, i])^7)((20736 (200843-34423 (Sqrt[21])))/(8575 ((τ[t])^7)))),τ'[t]->((-1+(Sqrt[21]))/2)(Subscript[H, i]),τ''[t]->0};

rep4={Derivative[5][H0][t]-> (((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^5)(-120((Subscript[H, i])/((τ[t])^6))),Derivative[5][G0][t]->((-7+(Sqrt[21]))/14)(((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^5)(-120((Subscript[H, i])/((τ[t])^6))),Derivative[4][H0][t]-> (((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^4)(24((Subscript[H, i])/((τ[t])^5))),Derivative[4][G0][t]->((-7+(Sqrt[21]))/14)(((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^4)(24((Subscript[H, i])/((τ[t])^5))),Derivative[3][H0][t]-> (((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^3)(-6((Subscript[H, i])/((τ[t])^4))),Derivative[3][G0][t]->((-7+(Sqrt[21]))/14)(((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^3)(-6((Subscript[H, i])/((τ[t])^4))),H0''[t]-> (((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^2)(2((Subscript[H, i])/((τ[t])^3))),G0''[t]->((-7+(Sqrt[21]))/14)(((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^2)(2((Subscript[H, i])/((τ[t])^3))),H0'[t]-> ((-1+(Sqrt[21]))/2)(Subscript[H, i])(-((Subscript[H, i])/((τ[t])^2))),G0'[t]->((-7+(Sqrt[21]))/14)((-1+(Sqrt[21]))/2)(Subscript[H, i])(-((Subscript[H, i])/((τ[t])^2))),H0[t]->((Subscript[H, i])/(τ[t])),G0[t]->((-7+(Sqrt[21]))/14)(((Subscript[H, i])/(τ[t]))),τ'[t]->((-1+(Sqrt[21]))/2)(Subscript[H, i]),τ''[t]->0};

rep5={Derivative[5][H1][t]-> (((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^5)(-55440 ((Subscript[H, i])^7)((13824 (477087-97732 (Sqrt[21])))/(8575 ((τ[t])^12)))),Derivative[5][G1][t]->((-7+(Sqrt[21]))/14)(((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^5)(-55440 ((Subscript[H, i])^7)((20736 (200843-34423 (Sqrt[21])))/(8575 ((τ[t])^12)))),Derivative[4][H1][t]-> (((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^4)(5040 ((Subscript[H, i])^7)((13824 (477087-97732 (Sqrt[21])))/(8575 ((τ[t])^11)))),Derivative[4][G1][t]->((-7+(Sqrt[21]))/14)(((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^4)(25040 ((Subscript[H, i])^7)((20736 (200843-34423 (Sqrt[21])))/(8575 ((τ[t])^11)))),Derivative[3][H1][t]-> (((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^3)(-504 ((Subscript[H, i])^7)((13824 (477087-97732 (Sqrt[21])))/(8575 ((τ[t])^10)))),Derivative[3][G1][t]->((-7+(Sqrt[21]))/14)(((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^3)(-504 ((Subscript[H, i])^7)((20736 (200843-34423 (Sqrt[21])))/(8575 ((τ[t])^10)))),H1''[t]-> (((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^2)(56 ((Subscript[H, i])^7)((13824 (477087-97732 (Sqrt[21])))/(8575 ((τ[t])^9)))),G1''[t]->((-7+(Sqrt[21]))/14)(((((-1+(Sqrt[21]))/2)(Subscript[H, i])))^2)(56 ((Subscript[H, i])^7)((20736 (200843-34423 (Sqrt[21])))/(8575 ((τ[t])^9)))),H1'[t]-> ((-1+(Sqrt[21]))/2)(Subscript[H, i])(-7 ((Subscript[H, i])^7)((13824 (477087-97732 (Sqrt[21])))/(8575 ((τ[t])^8)))),G1'[t]->((-7+(Sqrt[21]))/14)((-1+(Sqrt[21]))/2)(Subscript[H, i])(-7 ((Subscript[H, i])^7)((20736 (200843-34423 (Sqrt[21])))/(8575 ((τ[t])^8)))),H1[t]-> ((Subscript[H, i])^7)((13824 (477087-97732 (Sqrt[21])))/(8575 ((τ[t])^7))),G1[t]->((-7+(Sqrt[21]))/14)( ((Subscript[H, i])^7)((20736 (200843-34423 (Sqrt[21])))/(8575 ((τ[t])^7)))),τ'[t]->((-1+(Sqrt[21]))/2)(Subscript[H, i]),τ''[t]->0};

solsugra={Derivative[4][H0][t]->(3/2) (((1-(Sqrt[21])))^4) ((H0[t])^5),Derivative[3][H0][t]->(3/4) (((1-(Sqrt[21])))^3) ((H0[t])^4),H0''[t]->(1/2) (((1-(Sqrt[21])))^2) ((H0[t])^3),H0'[t]-> ((1-(Sqrt[21]))/2)((H0[t])^2),Derivative[4][G0][t]->(3/28) (((1-(Sqrt[21])))^4) (-7+(Sqrt[21])) ((H0[t])^5),Derivative[3][G0][t]->(3/56) (((1-(Sqrt[21])))^3) (-7+(Sqrt[21])) ((H0[t])^4),G0''[t]->(1/28) (((1-(Sqrt[21])))^2) (-7+(Sqrt[21])) ((H0[t])^3),G0'[t]->(1/28) (1-(Sqrt[21])) (-7+(Sqrt[21])) ((H0[t])^2),G0[t]->((-7+(Sqrt[21]))/14)H0[t]};

(* 2. Prescription eq.(17) *)

hp[t_]:=h0[t]+Γ h1[t];
H[t_]:=H0[t]+Γ H1[t];
G[t_]:=G0[t]+Γ G1[t];

(* 5. Tensor Perturbation eqs.(21), (25) *)

E1=Collect[FullSimplify[Normal[Series[Expand[E1Zero //. rep1 //. rep4 //. rep5 /. {(Subscript[H, i])->H0[t] τ[t]}],{Γ,0,1}]]],field]

(* 6. Derive eqs.(21),(26) *)

(* dt=a0dη, eq.(19) *)

Eη1=Collect[FullSimplify[Normal[Series[Expand[((((a0[t])^2)E1 /. Γ->0) /. {Derivative[2][h0][t]->(1/((a0[t])^2))(Derivative[2][h0][η]-ℋ0[η]Derivative[1][h0][η]),Derivative[1][h0][t]->(1/(a0[t]))Derivative[1][h0][η],h0[t]->h0[η],H0[t]->((ℋ0[η])/(a0[t])),a[t]->a0[t](1+Γ(OverBar[ a1])[η])})+(((a0[t])^2)Γ D[E1,Γ] /. {Derivative[4][h0][t]->(1/((a0[t])^4))(Derivative[4][h0][η]-6 ℋ0[η]Derivative[3][h0][η]+(5+2(Sqrt[21]))((ℋ0[η])^2)Derivative[2][h0][η]-((21+(Sqrt[21]))/2)((ℋ0[η])^3)Derivative[1][h0][η]),Derivative[3][h0][t]->(1/((a0[t])^3))(Derivative[3][h0][η]-3 ℋ0[η]Derivative[2][h0][η]+((1+(Sqrt[21]))/2)((ℋ0[η])^2)Derivative[1][h0][η]),Derivative[2][h0][t]->(1/((a0[t])^2))(Derivative[2][h0][η]-ℋ0[η]Derivative[1][h0][η]),Derivative[1][h0][t]->(1/(a0[t]))Derivative[1][h0][η],h0[t]->h0[η],H0[t]->(1/(a0[t]))ℋ0[η],h1[t]->h1[η],h1'[t]->(1/(a0[t]))h1'[η],h1''[t]->(1/((a0[t])^2))(Derivative[2][h1][η]-ℋ0[η]Derivative[1][h1][η]),a[t]->a0[t]})],{Γ,0,1}]]],{Γ,a0[t],h0[η],h0'[η],h0''[η],Derivative[3][h0][η],Derivative[4][h0][η],h1[η],h1'[η],h1''[η],k,ℋ0[η]}]

(* eq.(21), (26) *)

h0[η_]:=((a0[η])^((3-(Sqrt[21]))/4))U0[η];
h1[η_]:=((a0[η])^((3-(Sqrt[21]))/4))U1[η];

EC2=Collect[FullSimplify[Normal[Series[Expand[((a0[η])^(-((3-(Sqrt[21]))/4)))Eη1 //. {a0'[η]->a0[η]ℋ0[η],a0''[η]->a0[η]((ℋ0[η])^2)+a0[η]ℋ0'[η],Derivative[3][a0][η]->a0[η]((ℋ0[η])^3)+3a0[η]ℋ0[η]ℋ0'[η]+a0[η]ℋ0''[η],Derivative[4][a0][η]->a0[η] ((ℋ0[η])^4)+6 a0[η] ((ℋ0[η])^2) (ℋ0)'[η]+3 a0[η] (((ℋ0)'[η])^2)+4 a0[η] ℋ0[η] (ℋ0)''[η]+a0[η] Derivative[3][ℋ0][η],ℋ0'''[η]->(3/4) (((3-(Sqrt[21])))^3) ((ℋ0[η])^4),ℋ0''[η]->(1/2) (((3-(Sqrt[21])))^2) ((ℋ0[η])^3),ℋ0'[η]-> ((3-(Sqrt[21]))/2)((ℋ0[η])^2)}],{Γ,0,1}]]],{Γ,a0[t],U0[η],U0'[η],U0''[η],Derivative[3][U0][η],Derivative[4][U0][η],U1[η],U1'[η],U1''[η],k,ℋ0[η]}]

(* eq.(22) *)

DSolve[U0[η] ((k^2)+(1/(4(η^2))))+U0''[η]==0,U0[η],η]

(* eq.(27) *)

ℋ0[η_]:=((3+(Sqrt[21]))/6)(1/η);
(OverBar[a1])[η_]:=-((1+(Sqrt[21]))/60)((13824 (477087-97732 (Sqrt[21])))/8575)(((ℋ0[η])^6)/((a0[t])^6));

EC3=Collect[FullSimplify[Expand[(1/Γ)EC2 //. {U0''[η]->-((k^2)+(1/(4 (η^2)))) U0[η],Derivative[3][U0][η]->D[-((k^2)+(1/(4 (η^2)))) U0[η],η],Derivative[4][U0][η]->D[-((k^2)+(1/(4 (η^2)))) U0[η],{η,2}]} //. {U0[η]->c1(Sqrt[k η])BesselJ[0,k η]+c2(Sqrt[k η]) BesselY[0,k η],U0'[η]->D[c1(Sqrt[k η])BesselJ[0,k η]+c2(Sqrt[k η]) BesselY[0,k η],η],a0[t]->aE((((((Sqrt[21])-3)/2)aE HI η))^((3+(Sqrt[21]))/6))}]],{U1[η],U1'[η],U1''[η]}]

Simplify[(1/(231525 (aE^9) (HI^3) (η^11)))(2^(8+(Sqrt[21]))) (((-3+(Sqrt[21])))^(-(Sqrt[21]))) (9+2 (Sqrt[21])) (((aE HI η))^(-(Sqrt[21]))) (((k η))^(1/2)) -((2^8)/(3*5^2*7^3 (aE^6) (η^8))) (((((3+(Sqrt[21]))/6)))^(3+(Sqrt[21])))  (((aE HI η))^(-3-(Sqrt[21]))) (((k η))^(1/2))]

(* 7. eq.(27) *)

DSolve[((2^8)/(3*5^2*7^3 (aE^6) (η^8))) (((((3+(Sqrt[21]))/6)))^(3+(Sqrt[21])))  (((aE HI η))^(-3-(Sqrt[21]))) (((k η))^(1/2))((29265657+6318421 (Sqrt[21])) (((k η))^2) (c1 BesselJ[0,k η]+c2 BesselY[0,k η])-44100 (37+8 (Sqrt[21])) (k^4) (η^4) (c1 BesselJ[0,k η]+c2 BesselY[0,k η])+6 (4661457+1032913 (Sqrt[21])) (k η)(c1 BesselJ[1,k η]+c2 BesselY[1,k η])+73500 (217+47 (Sqrt[21])) (k^3) (η^3) (c1 BesselJ[1,k η]+c2 BesselY[1,k η]))+((k^2)+(1/(4 (η^2)))) U1[η]+U1''[η]==0,U1[η],η]

solU1=-((2^(6-(Sqrt[21]))) (3^(-(5/2)-(Sqrt[21]))) (((3+(Sqrt[21])))^(Sqrt[21])) π (((aE HI η))^(-(Sqrt[21]))) (-844527023 (Sqrt[3]) c1 (k^2) (η^2) (Sqrt[k η]) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] HypergeometricPFQ[{(1/2),-(7/2)-((Sqrt[21])/2)},{1,1,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-552875055 (Sqrt[7]) c1 (k^2) (η^2) (Sqrt[k η]) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] HypergeometricPFQ[{(1/2),-(7/2)-((Sqrt[21])/2)},{1,1,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-4890907273 (Sqrt[3]) c2 (k^2) (η^2) (Sqrt[k η]) BesselJ[0,k η] Gamma[(1/2) (-3-(Sqrt[21]))] Gamma[-(7/2)-((Sqrt[21])/2)] HypergeometricPFQ[{(1/2),-(7/2)-((Sqrt[21])/2)},{1,1,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-3201853227 (Sqrt[7]) c2 (k^2) (η^2) (Sqrt[k η]) BesselJ[0,k η] Gamma[(1/2) (-3-(Sqrt[21]))] Gamma[-(7/2)-((Sqrt[21])/2)] HypergeometricPFQ[{(1/2),-(7/2)-((Sqrt[21])/2)},{1,1,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+272890800 (Sqrt[3]) c2 (k^4) (η^4) (Sqrt[k η]) BesselJ[0,k η] ((Gamma[(1/2) (-5-(Sqrt[21]))])^2) HypergeometricPFQ[{(1/2),-(5/2)-((Sqrt[21])/2)},{1,1,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+178649100 (Sqrt[7]) c2 (k^4) (η^4) (Sqrt[k η]) BesselJ[0,k η] ((Gamma[(1/2) (-5-(Sqrt[21]))])^2) HypergeometricPFQ[{(1/2),-(5/2)-((Sqrt[21])/2)},{1,1,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+56955150 (Sqrt[3]) c1 (k^4) (η^4) (Sqrt[k η]) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] HypergeometricPFQ[{(1/2),-(5/2)-((Sqrt[21])/2)},{1,1,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+37286550 (Sqrt[7]) c1 (k^4) (η^4) (Sqrt[k η]) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] HypergeometricPFQ[{(1/2),-(5/2)-((Sqrt[21])/2)},{1,1,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-408839523 (Sqrt[3]) c1 (k^2) (η^2) (Sqrt[k η]) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] HypergeometricPFQ[{(3/2),-(7/2)-((Sqrt[21])/2)},{2,2,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-267646671 (Sqrt[7]) c1 (k^2) (η^2) (Sqrt[k η]) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] HypergeometricPFQ[{(3/2),-(7/2)-((Sqrt[21])/2)},{2,2,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-2367701679 (Sqrt[3]) c2 (k^2) (η^2) (Sqrt[k η]) BesselJ[0,k η] Gamma[(1/2) (-3-(Sqrt[21]))] Gamma[-(7/2)-((Sqrt[21])/2)] HypergeometricPFQ[{(3/2),-(7/2)-((Sqrt[21])/2)},{2,2,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-1550022633 (Sqrt[7]) c2 (k^2) (η^2) (Sqrt[k η]) BesselJ[0,k η] Gamma[(1/2) (-3-(Sqrt[21]))] Gamma[-(7/2)-((Sqrt[21])/2)] HypergeometricPFQ[{(3/2),-(7/2)-((Sqrt[21])/2)},{2,2,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-1334870250 (Sqrt[3]) c2 (k^4) (η^4) (Sqrt[k η]) BesselJ[0,k η] ((Gamma[(1/2) (-5-(Sqrt[21]))])^2) HypergeometricPFQ[{(3/2),-(5/2)-((Sqrt[21])/2)},{2,2,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-873878250 (Sqrt[7]) c2 (k^4) (η^4) (Sqrt[k η]) BesselJ[0,k η] ((Gamma[(1/2) (-5-(Sqrt[21]))])^2) HypergeometricPFQ[{(3/2),-(5/2)-((Sqrt[21])/2)},{2,2,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-278601750 (Sqrt[3]) c1 (k^4) (η^4) (Sqrt[k η]) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] HypergeometricPFQ[{(3/2),-(5/2)-((Sqrt[21])/2)},{2,2,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-182390250 (Sqrt[7]) c1 (k^4) (η^4) (Sqrt[k η]) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] HypergeometricPFQ[{(3/2),-(5/2)-((Sqrt[21])/2)},{2,2,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+545781600 c2 (k^4) (Sqrt[(3/π)]) (η^4) BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(1/4) (15+2 (Sqrt[21]))},{(3/4),(3/4)}},{{(1/4),(1/4),(1/4)},{(3/4),(1/4) (11+2 (Sqrt[21]))}},k η,(1/2)]+357298200 c2 (k^4) (Sqrt[(7/π)]) (η^4) BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(1/4) (15+2 (Sqrt[21]))},{(3/4),(3/4)}},{{(1/4),(1/4),(1/4)},{(3/4),(1/4) (11+2 (Sqrt[21]))}},k η,(1/2)]-5339481000 c2 (k^3) (Sqrt[(3/π)]) (η^3) BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(1/4) (17+2 (Sqrt[21]))},{(1/4),(1/4)}},{{-(1/4),-(1/4),(3/4)},{(1/4),(1/4) (13+2 (Sqrt[21]))}},k η,(1/2)]-3495513000 c2 (k^3) (Sqrt[(7/π)]) (η^3) BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(1/4) (17+2 (Sqrt[21]))},{(1/4),(1/4)}},{{-(1/4),-(1/4),(3/4)},{(1/4),(1/4) (13+2 (Sqrt[21]))}},k η,(1/2)]-9781814546 c2 (k^2) (Sqrt[(3/π)]) (η^2) BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(1/4) (19+2 (Sqrt[21]))},{(3/4),(3/4)}},{{(1/4),(1/4),(1/4)},{(3/4),(1/4) (15+2 (Sqrt[21]))}},k η,(1/2)]-6403706454 c2 (k^2) (Sqrt[(7/π)]) (η^2) BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(1/4) (19+2 (Sqrt[21]))},{(3/4),(3/4)}},{{(1/4),(1/4),(1/4)},{(3/4),(1/4) (15+2 (Sqrt[21]))}},k η,(1/2)]-9470806716 c2 k (Sqrt[(3/π)]) η BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(1/4) (21+2 (Sqrt[21]))},{(1/4),(1/4)}},{{-(1/4),-(1/4),(3/4)},{(1/4),(1/4) (17+2 (Sqrt[21]))}},k η,(1/2)]-6200090532 c2 k (Sqrt[(7/π)]) η BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(1/4) (21+2 (Sqrt[21]))},{(1/4),(1/4)}},{{-(1/4),-(1/4),(3/4)},{(1/4),(1/4) (17+2 (Sqrt[21]))}},k η,(1/2)]+2669740500 c2 (k^2) (Sqrt[(3/π)]) (η^2) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(3/4),(1/4) (19+2 (Sqrt[21]))},{-(1/4)}},{{(1/4),(5/4)},{-(1/4),(1/4),(1/4) (15+2 (Sqrt[21]))}},k η,(1/2)]+1747756500 c2 (k^2) (Sqrt[(7/π)]) (η^2) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(3/4),(1/4) (19+2 (Sqrt[21]))},{-(1/4)}},{{(1/4),(5/4)},{-(1/4),(1/4),(1/4) (15+2 (Sqrt[21]))}},k η,(1/2)]+4735403358 c2 (Sqrt[(3/π)]) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(3/4),(1/4) (23+2 (Sqrt[21]))},{-(1/4)}},{{(1/4),(5/4)},{-(1/4),(1/4),(1/4) (19+2 (Sqrt[21]))}},k η,(1/2)]+3100045266 c2 (Sqrt[(7/π)]) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(3/4),(1/4) (23+2 (Sqrt[21]))},{-(1/4)}},{{(1/4),(5/4)},{-(1/4),(1/4),(1/4) (19+2 (Sqrt[21]))}},k η,(1/2)]+272890800 c1 (k^3) (Sqrt[(3/π)]) (η^3) BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(5/4),(1/4) (17+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (13+2 (Sqrt[21]))}},k η,(1/2)]+178649100 c1 (k^3) (Sqrt[(7/π)]) (η^3) BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(5/4),(1/4) (17+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (13+2 (Sqrt[21]))}},k η,(1/2)]-272890800 c2 (k^3) (Sqrt[(3/π)]) (η^3) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(5/4),(1/4) (17+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (13+2 (Sqrt[21]))}},k η,(1/2)]-178649100 c2 (k^3) (Sqrt[(7/π)]) (η^3) BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(5/4),(1/4) (17+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (13+2 (Sqrt[21]))}},k η,(1/2)]-2669740500 c1 (k^2) (Sqrt[(3/π)]) (η^2) BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(5/4),(1/4) (19+2 (Sqrt[21]))},{(3/4)}},{{(5/4),(5/4)},{(1/4),(1/4),(1/4) (15+2 (Sqrt[21]))}},k η,(1/2)]-1747756500 c1 (k^2) (Sqrt[(7/π)]) (η^2) BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(5/4),(1/4) (19+2 (Sqrt[21]))},{(3/4)}},{{(5/4),(5/4)},{(1/4),(1/4),(1/4) (15+2 (Sqrt[21]))}},k η,(1/2)]-4890907273 c1 k (Sqrt[(3/π)]) η BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(5/4),(1/4) (21+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (17+2 (Sqrt[21]))}},k η,(1/2)]-3201853227 c1 k (Sqrt[(7/π)]) η BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(5/4),(1/4) (21+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (17+2 (Sqrt[21]))}},k η,(1/2)]+4890907273 c2 k (Sqrt[(3/π)]) η BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(5/4),(1/4) (21+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (17+2 (Sqrt[21]))}},k η,(1/2)]+3201853227 c2 k (Sqrt[(7/π)]) η BesselY[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(5/4),(1/4) (21+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (17+2 (Sqrt[21]))}},k η,(1/2)]-4735403358 c1 (Sqrt[(3/π)]) BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(5/4),(1/4) (23+2 (Sqrt[21]))},{(3/4)}},{{(5/4),(5/4)},{(1/4),(1/4),(1/4) (19+2 (Sqrt[21]))}},k η,(1/2)]-3100045266 c1 (Sqrt[(7/π)]) BesselJ[0,k η] Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))] MeijerG[{{(5/4),(1/4) (23+2 (Sqrt[21]))},{(3/4)}},{{(5/4),(5/4)},{(1/4),(1/4),(1/4) (19+2 (Sqrt[21]))}},k η,(1/2)]))/(8575 (14+3 (Sqrt[21])) (aE^9) (HI^3) (η^9) Gamma[(1/2) (-5-(Sqrt[21]))] Gamma[(1/2) (-3-(Sqrt[21]))]);

solU1simplify=-(((2^6)  ((((((Sqrt[21])-3)/2)))^9) π (HI^6))/(8575 (3^2)*7 ((((((Sqrt[21])-3)/2)aE HI η))^(9+(Sqrt[21]))))) (((14-3 (Sqrt[21]))(Sqrt[3]))/3)(-7 (((k η))^(5/2)) (c2 BesselJ[0,k η]-c1 BesselY[0,k η]) (1/14) (-23+5 (Sqrt[21])) ((19387374227 (Sqrt[3])+12692015805 (Sqrt[7])) HypergeometricPFQ[{(1/2),-(7/2)-((Sqrt[21])/2)},{1,1,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+3 (3128490419 (Sqrt[3])+2048077713 (Sqrt[7])) HypergeometricPFQ[{(3/2),-(7/2)-((Sqrt[21])/2)},{2,2,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-7350 (k^2) (η^2) (3 (59297 (Sqrt[3])+38819 (Sqrt[7])) HypergeometricPFQ[{(1/2),-(5/2)-((Sqrt[21])/2)},{1,1,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-70 (12431 (Sqrt[3])+8138 (Sqrt[7])) HypergeometricPFQ[{(3/2),-(5/2)-((Sqrt[21])/2)},{2,2,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]))+(1/(Sqrt[π]))(44100 (6188 (Sqrt[3])+4051 (Sqrt[7])) (k^3) (η^3) BesselJ[0,k η] (2 c2 k η MeijerG[{{(1/4) (15+2 (Sqrt[21]))},{(3/4),(3/4)}},{{(1/4),(1/4),(1/4)},{(3/4),(1/4) (11+2 (Sqrt[21]))}},k η,(1/2)]+c1 MeijerG[{{(5/4),(1/4) (17+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (13+2 (Sqrt[21]))}},k η,(1/2)])-514500 (5189 (Sqrt[3])+3397 (Sqrt[7])) c2 (k^2) (η^2) (2 k η BesselJ[0,k η] MeijerG[{{(1/4) (17+2 (Sqrt[21]))},{(1/4),(1/4)}},{{-(1/4),-(1/4),(3/4)},{(1/4),(1/4) (13+2 (Sqrt[21]))}},k η,(1/2)]-BesselY[0,k η] MeijerG[{{(3/4),(1/4) (19+2 (Sqrt[21]))},{-(1/4)}},{{(1/4),(5/4)},{-(1/4),(1/4),(1/4) (15+2 (Sqrt[21]))}},k η,(1/2)])+(4890907273 (Sqrt[3])+3201853227 (Sqrt[7])) k η (-2 c2 k η BesselJ[0,k η] MeijerG[{{(1/4) (19+2 (Sqrt[21]))},{(3/4),(3/4)}},{{(1/4),(1/4),(1/4)},{(3/4),(1/4) (15+2 (Sqrt[21]))}},k η,(1/2)]+(-c1 BesselJ[0,k η]+c2 BesselY[0,k η]) MeijerG[{{(5/4),(1/4) (21+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (17+2 (Sqrt[21]))}},k η,(1/2)])-14700 (k^2) (η^2) (3 (6188 (Sqrt[3])+4051 (Sqrt[7])) c2 k η BesselY[0,k η] MeijerG[{{(5/4),(1/4) (17+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (13+2 (Sqrt[21]))}},k η,(1/2)]+35 (5189 (Sqrt[3])+3397 (Sqrt[7])) c1 BesselJ[0,k η] MeijerG[{{(5/4),(1/4) (19+2 (Sqrt[21]))},{(3/4)}},{{(5/4),(5/4)},{(1/4),(1/4),(1/4) (15+2 (Sqrt[21]))}},k η,(1/2)])-6 (789233893 (Sqrt[3])+516674211 (Sqrt[7])) (2 c2 k η BesselJ[0,k η] MeijerG[{{(1/4) (21+2 (Sqrt[21]))},{(1/4),(1/4)}},{{-(1/4),-(1/4),(3/4)},{(1/4),(1/4) (17+2 (Sqrt[21]))}},k η,(1/2)]-c2 BesselY[0,k η] MeijerG[{{(3/4),(1/4) (23+2 (Sqrt[21]))},{-(1/4)}},{{(1/4),(5/4)},{-(1/4),(1/4),(1/4) (19+2 (Sqrt[21]))}},k η,(1/2)]+c1 BesselJ[0,k η] MeijerG[{{(5/4),(1/4) (23+2 (Sqrt[21]))},{(3/4)}},{{(5/4),(5/4)},{(1/4),(1/4),(1/4) (19+2 (Sqrt[21]))}},k η,(1/2)])));

FullSimplify[solU1-solU1simplify]

FullSimplify[Expand[((2^6)/(8575 (3^2)*7))((((((Sqrt[21])-3)/2)))^9)]]

solU1simplify2=(((2^6)  ((((((Sqrt[21])-3)/2)))^9) π (HI^6))/(8575 (3^2)*7 ((((((Sqrt[21])-3)/2)aE HI η))^(9+(Sqrt[21])))))( c2 ((((k η))^(5/2)) BesselJ[0,k η] (-(213002167+46502521 (Sqrt[21])) HypergeometricPFQ[{(1/2),-(7/2)-((Sqrt[21])/2)},{1,1,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-3 (34391077+7499743 (Sqrt[21])) HypergeometricPFQ[{(3/2),-(7/2)-((Sqrt[21])/2)},{2,2,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+51450 (279+61 (Sqrt[21])) (k^2) (η^2) HypergeometricPFQ[{(1/2),-(5/2)-((Sqrt[21])/2)},{1,1,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-85750 (819+179 (Sqrt[21])) (k^2) (η^2) HypergeometricPFQ[{(3/2),-(5/2)-((Sqrt[21])/2)},{2,2,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)])+(1/(Sqrt[π]))(BesselJ[0,k η](-205800 (669+146 (Sqrt[21])) (k^4) (η^4)  MeijerG[{{(1/4) (15+2 (Sqrt[21]))},{(3/4),(3/4)}},{{(1/4),(1/4),(1/4)},{(3/4),(1/4) (11+2 (Sqrt[21]))}},k η,(1/2)]+2*171500 (3927+857 (Sqrt[21])) (k^3) (η^3)  MeijerG[{{(1/4) (17+2 (Sqrt[21]))},{(1/4),(1/4)}},{{-(1/4),-(1/4),(3/4)},{(1/4),(1/4) (13+2 (Sqrt[21]))}},k η,(1/2)]+14 (176254865+38465701 (Sqrt[21])) (k^2) (η^2)  MeijerG[{{(1/4) (19+2 (Sqrt[21]))},{(3/4),(3/4)}},{{(1/4),(1/4),(1/4)},{(3/4),(1/4) (15+2 (Sqrt[21]))}},k η,(1/2)]+84 (28445153+6206377 (Sqrt[21])) k η  MeijerG[{{(1/4) (21+2 (Sqrt[21]))},{(1/4),(1/4)}},{{-(1/4),-(1/4),(3/4)},{(1/4),(1/4) (17+2 (Sqrt[21]))}},k η,(1/2)])+BesselY[0,k η](102900 (669+146 (Sqrt[21])) (k^3) (η^3)  MeijerG[{{(5/4),(1/4) (17+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (13+2 (Sqrt[21]))}},k η,(1/2)]-171500 (3927+857 (Sqrt[21])) (k^2) (η^2)  MeijerG[{{(3/4),(1/4) (19+2 (Sqrt[21]))},{-(1/4)}},{{(1/4),(5/4)},{-(1/4),(1/4),(1/4) (15+2 (Sqrt[21]))}},k η,(1/2)]-7 (176254865+38465701 (Sqrt[21])) k η MeijerG[{{(5/4),(1/4) (21+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (17+2 (Sqrt[21]))}},k η,(1/2)]-42 (28445153+6206377 (Sqrt[21]))  MeijerG[{{(3/4),(1/4) (23+2 (Sqrt[21]))},{-(1/4)}},{{(1/4),(5/4)},{-(1/4),(1/4),(1/4) (19+2 (Sqrt[21]))}},k η,(1/2)])))+c1 ((((k η))^(5/2)) BesselY[0,k η] ((213002167+46502521 (Sqrt[21])) HypergeometricPFQ[{(1/2),-(7/2)-((Sqrt[21])/2)},{1,1,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+3 (34391077+7499743 (Sqrt[21])) HypergeometricPFQ[{(3/2),-(7/2)-((Sqrt[21])/2)},{2,2,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-51450 (279+61 (Sqrt[21])) (k^2) (η^2) HypergeometricPFQ[{(1/2),-(5/2)-((Sqrt[21])/2)},{1,1,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+85750 (819+179 (Sqrt[21])) (k^2) (η^2) HypergeometricPFQ[{(3/2),-(5/2)-((Sqrt[21])/2)},{2,2,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)])+(1/(Sqrt[π]))BesselJ[0,k η](-102900 (669+146 (Sqrt[21])) (k^3) (η^3)  MeijerG[{{(5/4),(1/4) (17+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (13+2 (Sqrt[21]))}},k η,(1/2)]+171500 (3927+857 (Sqrt[21])) (k^2) (η^2)  MeijerG[{{(5/4),(1/4) (19+2 (Sqrt[21]))},{(3/4)}},{{(5/4),(5/4)},{(1/4),(1/4),(1/4) (15+2 (Sqrt[21]))}},k η,(1/2)]+7 (176254865+38465701 (Sqrt[21])) k η  MeijerG[{{(5/4),(1/4) (21+2 (Sqrt[21]))},{(1/4)}},{{(3/4),(3/4)},{(1/4),(3/4),(1/4) (17+2 (Sqrt[21]))}},k η,(1/2)]+42 (28445153+6206377 (Sqrt[21])) MeijerG[{{(5/4),(1/4) (23+2 (Sqrt[21]))},{(3/4)}},{{(5/4),(5/4)},{(1/4),(1/4),(1/4) (19+2 (Sqrt[21]))}},k η,(1/2)])));

FullSimplify[solU1simplify-solU1simplify2]

(* Set up : E1Zero=eq.(16) *)

E1Zero=-(1/(675 ((a[t])^4)))14336 (k^4) Γ hp[t] (92 ((G[t])^4)-322 ((G[t])^3) H[t]+33 ((H[t])^4)+((G[t])^2) (401 ((H[t])^2)+46 (G'[t]-H'[t]))-2 G[t] (102 ((H[t])^3)+59 H[t] (G'[t]-H'[t]))+72 ((H[t])^2) (G'[t]-H'[t])-13 (((G'[t]-H'[t]))^2))+hp''[t] (1+(1/3375)Γ (-6995968 ((G[t])^6)-97438208 ((G[t])^5) H[t]-23998464 ((H[t])^6)-45635072 ((G'[t])^3)-3584 ((G[t])^4) (30271 ((H[t])^2)+34969 G'[t]-23579 H'[t])+21504 ((H[t])^4) (3254 G'[t]-6999 H'[t])+54114816 ((G'[t])^2) H'[t]+28675584 G'[t] ((H'[t])^2)-37155328 ((H'[t])^3)-11038720 ((G''[t])^2)+107520 ((H[t])^3) (523 G''[t]-595 H''[t])+22077440 G''[t] H''[t]-11038720 ((H''[t])^2)+3584 ((G[t])^3) (112291 ((H[t])^3)-4 H[t] (40824 G'[t]-34579 H'[t])-11110 G''[t]+10190 H''[t])-501760 H[t] (G'[t] (195 G''[t]-217 H''[t])+H'[t] (-239 G''[t]+261 H''[t]))-11038720 G'[t] Derivative[3][G][t]+11038720 H'[t] Derivative[3][G][t]-3584 G[t] (24228 ((H[t])^5)-6 ((H[t])^3) (21353 G'[t]-21823 H'[t])+((H[t])^2) (-25270 G''[t]+21870 H''[t])+140 (G'[t] (333 G''[t]-311 H''[t])+H'[t] (-289 G''[t]+267 H''[t]))+H[t] (130037 ((G'[t])^2)-262674 G'[t] H'[t]+126477 ((H'[t])^2)+1240 (Derivative[3][G][t]-Derivative[3][H][t])))-3584 ((H[t])^2) (7367 ((G'[t])^2)-68874 G'[t] H'[t]+64587 ((H'[t])^2)-2160 (Derivative[3][G][t]-Derivative[3][H][t]))-3584 ((G[t])^2) (21957 ((H[t])^4)+98216 ((G'[t])^2)-139692 G'[t] H'[t]+44556 ((H'[t])^2)-((H[t])^2) (50623 G'[t]+11037 H'[t])+10 H[t] (2985 G''[t]-2953 H''[t])+920 Derivative[3][G][t]-920 Derivative[3][H][t])+11038720 G'[t] Derivative[3][H][t]-11038720 H'[t] Derivative[3][H][t]))+hp'[t] (((23625 G[t]+10125 H[t])/3375)+(1/3375)Γ (200704 ((G[t])^7)+294443520 ((G[t])^6) H[t]-27869184 G[t] ((H[t])^6)-8128512 ((H[t])^7)+68300288 G[t] ((G'[t])^3)-86216704 H[t] ((G'[t])^3)+129024 G[t] ((H[t])^4) (1548 G'[t]-3077 H'[t])+193536 ((H[t])^5) (86 G'[t]-347 H'[t])-249518080 G[t] ((G'[t])^2) H'[t]+104570368 H[t] ((G'[t])^2) H'[t]-90840064 G[t] ((H'[t])^3)-89940480 H[t] ((H'[t])^3)-3584 ((G[t])^5) (286878 ((H[t])^2)-88465 G'[t]+63454 H'[t])+7239680 G[t] ((G''[t])^2)-18278400 H[t] ((G''[t])^2)+3584 ((G[t])^4) (243124 ((H[t])^3)+H[t] (-83314 G'[t]+3967 H'[t])+26141 G''[t]-24251 H''[t])+21504 G[t] ((H[t])^3) (7333 G''[t]-6963 H''[t])+21504 ((H[t])^4) (1429 G''[t]-1914 H''[t])-14479360 G[t] G''[t] H''[t]+36556800 H[t] G''[t] H''[t]+7239680 G[t] ((H''[t])^2)-18278400 H[t] ((H''[t])^2)-43008 ((H[t])^2) (G'[t] (1757 G''[t]-2362 H''[t])+H'[t] (-2967 G''[t]+3572 H''[t]))+3584 (G'[t]-H'[t]) (G'[t] (1681 G''[t]-3221 H''[t])+H'[t] (-7841 G''[t]+9381 H''[t]))-7168 G[t] H[t] (G'[t] (43227 G''[t]-39047 H''[t])+H'[t] (-34867 G''[t]+30687 H''[t]))-7239680 G[t] H'[t] Derivative[3][G][t]+3584 ((G[t])^3) (33174 ((H[t])^4)+((H[t])^2) (-369471 G'[t]+369900 H'[t])+H[t] (-52736 G''[t]+45356 H''[t])+2 (59996 ((G'[t])^2)-99948 G'[t] H'[t]+40042 ((H'[t])^2)+965 (Derivative[3][G][t]-Derivative[3][H][t])))-3584 ((G[t])^2) (61587 ((H[t])^5)+((H[t])^3) (-303948 G'[t]+180903 H'[t])+2 H'[t] (18789 G''[t]-18239 H''[t])+3 ((H[t])^2) (8659 G''[t]-10719 H''[t])+G'[t] (-39778 G''[t]+38678 H''[t])+H[t] (213335 ((G'[t])^2)-241268 G'[t] H'[t]+37713 ((H'[t])^2)+4690 (Derivative[3][G][t]-Derivative[3][H][t])))+21504 H[t] G'[t] (3329 ((H'[t])^2)-850 (Derivative[3][G][t]-Derivative[3][H][t]))+10752 ((H[t])^3) (5759 ((G'[t])^2)+7820 G'[t] H'[t]-16719 ((H'[t])^2)+750 (Derivative[3][G][t]-Derivative[3][H][t]))+3584 G[t] G'[t] (75909 ((H'[t])^2)+2020 (Derivative[3][G][t]-Derivative[3][H][t]))+18278400 H[t] H'[t] (Derivative[3][G][t]-Derivative[3][H][t])+7239680 G[t] H'[t] Derivative[3][H][t]-21504 G[t] ((H[t])^2) (20689 ((G'[t])^2)-55928 G'[t] H'[t]+32069 ((H'[t])^2)-85 Derivative[3][G][t]+85 Derivative[3][H][t])))+(1/((a[t])^2))(k^2) ((1/10125)Γ hp'[t] (27847680 ((G[t])^5)+92789760 ((G[t])^4) H[t]-347827200 ((G[t])^3) ((H[t])^2)+291701760 ((G[t])^2) ((H[t])^3)-50319360 G[t] ((H[t])^4)-14192640 ((H[t])^5)+215040 ((G[t])^3) (1005 G'[t]-872 H'[t])-215040 ((G[t])^2) H[t] (873 G'[t]-571 H'[t])+2580480 ((H[t])^3) (22 G'[t]-25 H'[t])+215040 G[t] ((H[t])^2) (-396 G'[t]+601 H'[t])+107520 H[t] (-241 ((G'[t])^2)+288 G'[t] H'[t]-47 ((H'[t])^2))+107520 G[t] (1211 ((G'[t])^2)-2228 G'[t] H'[t]+1017 ((H'[t])^2))+28600320 ((G[t])^2) (G''[t]-H''[t])-36341760 G[t] H[t] (G''[t]-H''[t])+7741440 ((H[t])^2) (G''[t]-H''[t])+20858880 (G'[t]-H'[t]) (G''[t]-H''[t]))+(1/10125)Γ (3978240 ((G[t])^4)+12687360 ((G[t])^3) H[t]-51502080 ((G[t])^2) ((H[t])^2)+49029120 G[t] ((H[t])^3)-14192640 ((H[t])^4)+28600320 ((G[t])^2) (G'[t]-H'[t])-36341760 G[t] H[t] (G'[t]-H'[t])+7741440 ((H[t])^2) (G'[t]-H'[t])+10429440 (((G'[t]-H'[t]))^2)) hp''[t]+hp[t] (1+(1/10125)Γ (587776 ((G[t])^6)-559286784 ((G[t])^5) H[t]+6064128 ((H[t])^6)+28962304 ((G'[t])^3)-24202752 ((G'[t])^2) H'[t]-38481408 G'[t] ((H'[t])^2)+33721856 ((H'[t])^3)+64512 ((H[t])^4) (1644 G'[t]+181 H'[t])+10752 ((G[t])^4) (184029 ((H[t])^2)-56519 G'[t]+42059 H'[t])+11397120 ((G''[t])^2)-3584 ((G[t])^3) (634247 ((H[t])^3)+H[t] (-383772 G'[t]+227652 H'[t])+3030 (15 G''[t]-14 H''[t]))-322560 ((H[t])^3) (119 G''[t]-188 H''[t])-22794240 G''[t] H''[t]+11397120 ((H''[t])^2)+215040 H[t] (G'[t] (436 G''[t]-489 H''[t])+H'[t] (-542 G''[t]+595 H''[t]))+11397120 G'[t] Derivative[3][G][t]-11397120 H'[t] Derivative[3][G][t]-10752 G[t] (318 ((H[t])^5)+((H[t])^3) (80102 G'[t]-81322 H'[t])+10 ((H[t])^2) (2497 G''[t]-1982 H''[t])-20 (G'[t] (518 G''[t]-465 H''[t])+H'[t] (-412 G''[t]+359 H''[t]))+H[t] (-61683 ((G'[t])^2)+102386 G'[t] H'[t]-38583 ((H'[t])^2)-3080 (Derivative[3][G][t]-Derivative[3][H][t])))+10752 ((G[t])^2) (79103 ((H[t])^4)-((H[t])^2) (1167 G'[t]+48583 H'[t])+10 H[t] (4369 G''[t]-3960 H''[t])-2 (2503 ((G'[t])^2)-6651 G'[t] H'[t]+3618 ((H'[t])^2)+505 (Derivative[3][G][t]-Derivative[3][H][t])))-10752 ((H[t])^2) (10037 ((G'[t])^2)+4196 G'[t] H'[t]-15293 ((H'[t])^2)+2070 (Derivative[3][G][t]-Derivative[3][H][t]))-11397120 G'[t] Derivative[3][H][t]+11397120 H'[t] Derivative[3][H][t])))-(28672/675) Γ (14 ((G[t])^5)+272 ((G[t])^4) H[t]+99 ((H[t])^5)+((G[t])^3) (-229 ((H[t])^2)+330 G'[t]-284 H'[t])+((H[t])^3) (-348 G'[t]+456 H'[t])+H[t] (293 ((G'[t])^2)-740 G'[t] H'[t]+447 ((H'[t])^2))+((G[t])^2) (-315 ((H[t])^3)+H[t] (686 G'[t]-670 H'[t])+46 (G''[t]-H''[t]))+G[t] (159 ((H[t])^4)+631 ((G'[t])^2)-1108 G'[t] H'[t]+477 ((H'[t])^2)+((H[t])^2) (-668 G'[t]+498 H'[t])+62 H[t] (G''[t]-H''[t]))-108 ((H[t])^2) (G''[t]-H''[t])+154 (G'[t]-H'[t]) (G''[t]-H''[t])) Derivative[3][hp][t]-(14336/675) Γ (2 ((G[t])^4)+38 ((G[t])^3) H[t]+33 ((H[t])^4)+((G[t])^2) (-49 ((H[t])^2)+46 (G'[t]-H'[t]))+G[t] (-24 ((H[t])^3)+62 H[t] (G'[t]-H'[t]))-108 ((H[t])^2) (G'[t]-H'[t])+77 (((G'[t]-H'[t]))^2)) Derivative[4][hp][t];

Expand[E1-E1Zero]

(* Check with tex *)

(* A0 *)

A0=Collect[D[E1Zero,Γ] /. {hp[t]->1,Derivative[1][hp][t]->0,Derivative[2][hp][t]->0,Derivative[3][hp][t]->0,Derivative[4][hp][t]->0},{k,a[t]}] /. {f_[t_]->f}

A0tex = ((3584k^2)/(10125a^2))(164 G^6-156051 G^5 H+552087 G^4 H^2+126177 G^4H'-169557 G'  G^4-634247 G^3 H^3+42420 G^3 H'' +383772 G'  G^3 H-227652 G^3 H H'-45450 G^3 G'' +237309 G^2 H^4-3501 G'  G^2 H^2-145749 G^2 H^2 H' +131070 G^2 H G'' -118800 G^2 H H'' +3030 G^2 Derivative[3][H] -21708 G^2 H' ^2+39906 G'  G^2 H' -3030 G^2 Derivative[3][G] -15018 G' ^2 G^2-954 G H^5+29592 G'  H^4-10710 H^3 G'' -240306 G'  G H^3+243966 G H^3 H' -74910 G H^2 G'' +59460 G H^2 H'' -6210 H^2 Derivative[3][G] -30111 G' ^2 H^2-12588 G'  H^2 H' -24720 G H'  G'' -27900 G'  G H'' +21540 G H'  H'' +26160 G'  H G'' -32520 H H'  G'' -29340 G'  H H'' -6360 G''  H'' +9240 G H Derivative[3][G] -9240 G H Derivative[3][H] -3180 H'  Derivative[3][G] -3180 G'  Derivative[3][H] +185049 G' ^2 G H+115749 G H H' ^2-307158 G'  G H H' -10737 G'  H' ^2-6753 G' ^2 H' +31080 G'  G G'' +3180 G'' ^2+3180 G'  Derivative[3][G] +8081 G' ^3+1692 H^6+3258 H^4 H' +16920 H^3 H'' +6210 H^2 Derivative[3][H] +45879 H^2 H' ^2+3180 H'' ^2+35700 H H'  H'' +3180 H'  Derivative[3][H] +9409 H' ^3)- ((14336k^4)/(675a^4))(92 G^4-322 G^3 H+401 G^2 H^2-46 G^2 H' +46 G'  G^2-204 G H^3+72 G'  H^2-118 G'  G H+118 G H H' +26 G'  H' -13 G' ^2+33 H^4-72 H^2 H' -13 H' ^2)

Expand[A0-A0tex]

(* A1 *)

A1=Collect[D[E1Zero,Γ] /. {hp[t]->0,Derivative[1][hp][t]->1,Derivative[2][hp][t]->0,Derivative[3][hp][t]->0,Derivative[4][hp][t]->0},{k,a[t]}] /. {f_[t_]->f}

A1tex= (3584/3375)(56 G^7+82155 H G^6-286878 H^2 G^5+88465 G'  G^5-63454 H'  G^5+243124 H^3 G^4 -83314 H G'  G^4+26141 G''  G^4+3967 H H'  G^4-24251 H''  G^4+33174 H^4 G^3+119992 G' ^2 G^3 +80084 H' ^2 G^3-369471 H^2 G'  G^3-52736 H G''  G^3+369900 H^2 H'  G^3-199896 G'  H'  G^3 +45356 H H''  G^3+1930 Derivative[3][G]  G^3-1930 Derivative[3][H]  G^3-61587 H^5 G^2-213335 H G' ^2 G^2 -37713 H H' ^2 G^2+303948 H^3 G'  G^2-25977 H^2 G''  G^2+39778 G'  G''  G^2-180903 H^3 H'  G^2 +241268 H G'  H'  G^2-37578 G''  H'  G^2+32157 H^2 H''  G^2-38678 G'  H''  G^2+36478 H'  H''  G^2 -4690 H Derivative[3][G]  G^2+4690 H Derivative[3][H]  G^2-7776 H^6 G+19057 G' ^3 G-25346 H' ^3 G-124134 H^2 G' ^2 G +2020 G'' ^2 G-192414 H^2 H' ^2 G+75909 G'  H' ^2 G+2020 H'' ^2 G+55728 H^4 G'  G +43998 H^3 G''  G-86454 H G'  G''  G-110772 H^4 H'  G-69620 G' ^2 H'  G+335568 H^2 G'  H'  G +69734 H G''  H'  G-41778 H^3 H''  G+78094 H G'  H''  G-4040 G''  H''  G-61374 H H'  H''  G +510 H^2 Derivative[3][G]  G+2020 G'  Derivative[3][G]  G-2020 H'  Derivative[3][G]  G-510 H^2 Derivative[3][H]  G-2020 G'  Derivative[3][H]  G+2020 H'  Derivative[3][H]  G-2268 H^7-24056 H G' ^3-25095 H H' ^3+17277 H^3 G' ^2-5100 H G'' ^2-50157 H^3 H' ^2 +19974 H G'  H' ^2+7841 G''  H' ^2-5100 H H'' ^2+4644 H^5 G' +8574 H^4 G'' +1681 G' ^2 G''  -21084 H^2 G'  G'' -18738 H^5 H' +29177 H G' ^2 H' +23460 H^3 G'  H' +35604 H^2 G''  H'  -9522 G'  G''  H' -11484 H^4 H'' -3221 G' ^2 H'' -9381 H' ^2 H'' +28344 H^2 G'  H'' +10200 H G''  H''  -42864 H^2 H'  H'' +12602 G'  H'  H'' +2250 H^3 Derivative[3][G] -5100 H G'  Derivative[3][G] +5100 H H'  Derivative[3][G] -2250 H^3 Derivative[3][H]  +5100 H G'  Derivative[3][H] -5100 H H'  Derivative[3][H] )+ ((7168k^2)/(675a^2))(259 G^5+863 G^4 H-3235 G^3 H^2-1744 G^3 H'  +2010 G'  G^3+2713 G^2 H^3-266 G^2 H'' -1746 G'  G^2 H+1142 G^2 H H' +266 G^2 G'' -468 G H^4 +528 G'  H^3+72 H^2 G'' -792 G'  G H^2+1202 G H^2 H' -338 G H G'' +338 G H H'' -194 H'  G''  -194 G'  H'' +1017 G H' ^2-2228 G'  G H' -241 G' ^2 H+288 G'  H H' +194 G'  G'' +1211 G' ^2 G -132 H^5-600 H^3 H' -72 H^2 H'' +194 H'  H'' -47 H H' ^2)

Expand[A1-A1tex]

(* A2 *)

A2=Collect[D[E1Zero,Γ] /. {hp[t]->0,Derivative[1][hp][t]->0,Derivative[2][hp][t]->1,Derivative[3][hp][t]->0,Derivative[4][hp][t]->0},{k,a[t]}] /. {f_[t_]->f}

A2tex=- (3584/3375)(1952 G^6+27187 G^5 H+30271 G^4 H^2-23579 G^4 H' +34969 G'  G^4-112291 G^3 H^3 -10190 G^3 H'' +163296 G'  G^3 H-138316 G^3 H H' +11110 G^3 G'' +21957 G^2 H^4 -50623 G'  G^2 H^2-11037 G^2 H^2 H' +29850 G^2 H G'' -29530 G^2 H H'' -920 G^2 Derivative[3][H]  +44556 G^2 H' ^2-139692 G'  G^2 H' +920 G^2 Derivative[3][G] +98216 G' ^2 G^2+24228 G H^5-19524 G'  H^4 -15690 H^3 G'' -128118 G'  G H^3+130938 G H^3 H' -25270 G H^2 G'' +21870 G H^2 H''-2160 H^2 Derivative[3][G] +7367 G' ^2 H^2-68874 G'  H^2 H' -40460 G H'  G'' -43540 G'  G H'' +37380 G H'  H''  +27300 G'  H G'' -33460 H H'  G'' -30380 G'  H H'' -6160 G''  H'' +1240 G H Derivative[3][G] -1240 G H Derivative[3][H]  -3080 H'  Derivative[3][G] -3080 G'  Derivative[3][H] +130037 G' ^2 G H+126477 G H H' ^2-262674 G'  G H H' -8001 G'  H' ^2 -15099 G' ^2 H' +46620 G'  G G'' +3080 G'' ^2+3080 G'  Derivative[3][G] +12733 G' ^3+6696 H^6+41994 H^4 H'  +17850 H^3 H'' +2160 H^2 Derivative[3][H] +64587 H^2 H' ^2+3080 H'' ^2+36540 H H'  H'' +3080 H'  Derivative[3][H]  +10367 H' ^3 )+ ((7168k^2)/(675a^2))(37 G^4+118 G^3 H-479 G^2 H^2-266 G^2 H' +266 G'  G^2+456 G H^3 +72 G'  H^2-338 G'  G H+338 G H H' -194 G'  H' +97 G' ^2-132 H^4-72 H^2 H' +97 H' ^2 )

Expand[A2-A2tex]

(* A3 *)

A3=Collect[D[E1Zero,Γ] /. {hp[t]->0,Derivative[1][hp][t]->0,Derivative[2][hp][t]->0,Derivative[3][hp][t]->1,Derivative[4][hp][t]->0},{k,a[t]}] /. {f_[t_]->f}

A3tex =- (28672/675)(14 G^5+272 G^4 H-229 G^3 H^2-284 G^3 H' +330 G'  G^3-315 G^2 H^3 -46 G^2 H'' +686 G'  G^2 H-670 G^2 H H' +46 G^2 G'' +159 G H^4-348 G'  H^3 -108 H^2 G'' -668 G'  G H^2+498 G H^2 H' +62 G H G'' -62 G H H'' -154 H'  G''-154 G'  H'' +477 G H' ^2-1108 G'  G H' +293 G' ^2 H-740 G'  H H' +154 G'  G''  +631 G' ^2 G+99 H^5+456 H^3 H' +108 H^2 H'' +154 H'  H'' +447 H H' ^2 )

Expand[A3-A3tex]

(* A3 *)

A4=Collect[D[E1Zero,Γ] /. {hp[t]->0,Derivative[1][hp][t]->0,Derivative[2][hp][t]->0,Derivative[3][hp][t]->0,Derivative[4][hp][t]->1},{k,a[t]}] /. {f_[t_]->f}

A4tex=-(14336/675)(2 G^4+38 G^3 H-49 G^2 H^2-46 G^2 H' +46 G'  G^2-24 G H^3-108 G'  H^2 +62 G'  G H-62 G H H' -154 G'  H' +77 G' ^2+33 H^4+108 H^2 H' +77 H' ^2 )

Expand[A4-A4tex]

Expand[(D[E1Zero,Γ] /. {f_[t_]->f})-A0tex hp-A1tex hp'-A2tex hp''-A3tex Derivative[3][hp]-A4tex Derivative[4][hp]]