G:=1/(8*Pi^2*σ[a,b,c,d,p,q]);
s:=σ[a,b,c,d,p,q];

sa:=σa[a,b,c,d,p,q];
sb:=σb[a,b,c,d,p,q];
sc:=σc[a,b,c,d,p,q];
sd:=σd[a,b,c,d,p,q];
sp:=σp[a,b,c,d,p,q];
sq:=σq[a,b,c,d,p,q];

Format[σ[a,b,c,d,p,q]]:="s";
Format[σa[a,b,c,d,p,q]]:="sa";
Format[σb[a,b,c,d,p,q]]:="sb";
Format[σc[a,b,c,d,p,q]]:="sc";
Format[σd[a,b,c,d,p,q]]:="sd";
Format[σp[a,b,c,d,p,q]]:="sp";
Format[σq[a,b,c,d,p,q]]:="sq";

Derivative[1,0,0,0,0,0][σ][a_,b_,c_,d_,p_,q_]:=sa;
Derivative[0,1,0,0,0,0][σ][a_,b_,c_,d_,p_,q_]:=sb;
Derivative[0,0,1,0,0,0][σ][a_,b_,c_,d_,p_,q_]:=sc;
Derivative[0,0,0,1,0,0][σ][a_,b_,c_,d_,p_,q_]:=sd;
Derivative[0,0,0,0,1,0][σ][a_,b_,c_,d_,p_,q_]:=sp;
Derivative[0,0,0,0,0,1][σ][a_,b_,c_,d_,p_,q_]:=sq;

Derivative[0,1,0,0,0,0][σa][a_,b_,c_,d_,p_,q_]:=gab;
Derivative[0,0,1,0,0,0][σa][a_,b_,c_,d_,p_,q_]:=gac;
Derivative[0,0,0,1,0,0][σa][a_,b_,c_,d_,p_,q_]:=gad;
Derivative[0,0,0,0,1,0][σa][a_,b_,c_,d_,p_,q_]:=gap;
Derivative[0,0,0,0,0,1][σa][a_,b_,c_,d_,p_,q_]:=gaq;
Derivative[1,0,0,0,0,0][σb][a_,b_,c_,d_,p_,q_]:=gab;
Derivative[0,0,1,0,0,0][σb][a_,b_,c_,d_,p_,q_]:=gbc;
Derivative[0,0,0,1,0,0][σb][a_,b_,c_,d_,p_,q_]:=gbd;
Derivative[0,0,0,0,1,0][σb][a_,b_,c_,d_,p_,q_]:=gbp;
Derivative[0,0,0,0,0,1][σb][a_,b_,c_,d_,p_,q_]:=gbq;
Derivative[1,0,0,0,0,0][σc][a_,b_,c_,d_,p_,q_]:=gac;
Derivative[0,1,0,0,0,0][σc][a_,b_,c_,d_,p_,q_]:=gbc;
Derivative[0,0,0,1,0,0][σc][a_,b_,c_,d_,p_,q_]:=gcd;
Derivative[0,0,0,0,1,0][σc][a_,b_,c_,d_,p_,q_]:=gcp;
Derivative[0,0,0,0,0,1][σc][a_,b_,c_,d_,p_,q_]:=gcq;
Derivative[1,0,0,0,0,0][σd][a_,b_,c_,d_,p_,q_]:=gad;
Derivative[0,1,0,0,0,0][σd][a_,b_,c_,d_,p_,q_]:=gbd;
Derivative[0,0,1,0,0,0][σd][a_,b_,c_,d_,p_,q_]:=gcd;
Derivative[0,0,0,0,1,0][σd][a_,b_,c_,d_,p_,q_]:=gdp;
Derivative[0,0,0,0,0,1][σd][a_,b_,c_,d_,p_,q_]:=gdq;
Derivative[1,0,0,0,0,0][σp][a_,b_,c_,d_,p_,q_]:=gap;
Derivative[0,1,0,0,0,0][σp][a_,b_,c_,d_,p_,q_]:=gbp;
Derivative[0,0,1,0,0,0][σp][a_,b_,c_,d_,p_,q_]:=gcp;
Derivative[0,0,0,1,0,0][σp][a_,b_,c_,d_,p_,q_]:=gdp;
Derivative[0,0,0,0,1,0][σp][a_,b_,c_,d_,p_,q_]:=4;
Derivative[0,0,0,0,0,1][σp][a_,b_,c_,d_,p_,q_]:=gpq;
Derivative[1,0,0,0,0,0][σq][a_,b_,c_,d_,p_,q_]:=gaq;
Derivative[0,1,0,0,0,0][σq][a_,b_,c_,d_,p_,q_]:=gbq;
Derivative[0,0,1,0,0,0][σq][a_,b_,c_,d_,p_,q_]:=gcq;
Derivative[0,0,0,1,0,0][σq][a_,b_,c_,d_,p_,q_]:=gdq;
Derivative[0,0,0,0,1,0][σq][a_,b_,c_,d_,p_,q_]:=gpq;
Derivative[0,0,0,0,0,1][σq][a_,b_,c_,d_,p_,q_]:=4;

prules={gap*gbp->gab,gap*sp->sa,gbp*sp->sb,sp^2->2s,gcp*gdp->gcd,gcp*sp->sc,gdp*sp->sd,gpq^2->4,gpq*sp->sq};
qrules={gaq*gbq->gab,gaq*sq->sa,gbq*sq->sb,sq^2->2s,gcq*gdq->gcd,gcq*sq->sc,gdq*sq->sd};

K11=4*(D[G,c,b]D[G,a,d]+D[G,c,a]D[G,b,d])//Simplify

K12=D[G,a,b]D[G,c,d]//Simplify

K13=G*D[G,a,b,c,d]//Simplify

K14=-2*(D[G,a]D[G,b,c,d]+D[G,b]D[G,a,c,d]+D[G,c]D[G,b,a,d]+D[G,d]D[G,b,c,a])//Simplify

K1s=K11+K12+K13+K14//Simplify

K21=-8*Expand[D[G,a,p]D[G,b,p]]/.prules//Simplify

K22=4*Expand[D[G,p]D[G,a,b,p]]/.prules//Simplify

K2s=K21+K22//Simplify

K31=-8*Expand[D[G,c,q]D[G,d,q]]/.qrules//Simplify

K32=4*Expand[D[G,q]D[G,c,d,q]]/.qrules//Simplify

K3s=K31+K32//Simplify

K41=2*Expand[D[G,p,q]*D[G,p,q]]/.prules/.qrules//Simplify

K4s=K41//Simplify

Ksum=K1s/9+gcd*K2s/36+gab*K3s/36+gab*gcd*K4s/36//Expand

Ksum/.{s->(-t^2+r^2)/2}/.{gab->1,gac->-1,gad->-1,gbc->-1,gbd->-1,gcd->1}/.{sa->x,sb->y,sc->-x,sd->-y}//Simplify

Ksum2=Sum[Coefficient[Ksum,s,n]*(s)^n/.{s->tr/2}//Simplify,{n,-6,-4}]

Ksum3=Ksum2/.{gab->-1,gac->1,gad->1,gbc->1,gbd->1,gcd->-1}/.{sa->-t,sb->-t,sc->t,sd->t}

%//Simplify