Clear[coord,metric,inversemetric,affine,t,x,y,z]

r := Sqrt[x^2 + y^2+z^2]

n:=4

coord:={t,x,y,z}

metric:={{-B[r],0,0,0},
{0,A[r]+x^2*(-A[r]+C[r])/r^2,x*y*(-A[r]+C[r])/r^2,x*z*(-A[r]+C[r])/r^2},
{0,y*x*(-A[r]+C[r])/r^2,A[r]+y^2*(-A[r]+C[r])/r^2,y*z*(-A[r]+C[r])/r^2},
{0,z*x*(-A[r]+C[r])/r^2,z*y*(-A[r]+C[r])/r^2,A[r]+z^2*(-A[r]+C[r])/r^2}}

CC[r]:=A[r]+r^2*G[r]

rule1 = {A[(Sqrt[(x^2)+(y^2)+(z^2)])]+((x^2)+(y^2)+(z^2)) G[(Sqrt[(x^2)+(y^2)+(z^2)])]-> C[r]};

Simplify[Simplify[A[Sqrt[x^2+y^2+z^2]]+Sqrt[x^2+y^2+z^2]^2*G[Sqrt[x^2+y^2+z^2]]/.rule1]]

metric//MatrixForm

DET=Simplify[Simplify[Simplify[Simplify[Simplify[Simplify[Det[metric]]/.rule2]/.rule3]/.rule4]/.rule5]/.rule6]

inversemetric:=Simplify[Inverse[metric]]

inversemetric//MatrixForm

Simplify[Simplify[Det[inversemetric]/.rule1]/.rule1]

Calculating the Christoffel symbols of the second kind:

affine:=affine=Simplify[Table[(1/2) Sum[inversemetric[[Mu,Rho]] (D[metric[[Rho,Nu]],coord[[Lambda]]]+D[metric[[Rho,Lambda]],coord[[Nu]]]-D[metric[[Nu,Lambda]],coord[[Rho]]]),{Rho,1,n}],{Nu,1,n},{Lambda,1,n},{Mu,1,n}]]

listaffine:=Table[If[UnsameQ[affine[[Nu,Lambda,Mu]],0],{Style[Subsuperscript[Γ,Row[{coord[[Nu]],coord[[Lambda]]}],coord[[Mu]]],18],"=",Style[affine[[Nu,Lambda,Mu]],14]}],{Lambda,1,n},{Nu,1,Lambda},{Mu,1,n}]

Simplify[TableForm[Partition[DeleteCases[Flatten[listaffine],Null],3],TableSpacing->{1,2}]/.rule1]

Defining the Riemann tensor.

riemann:=riemann=Table[D[affine[[Nu,Sigma,Mu]],coord[[Rho]]]-D[affine[[Nu,Rho,Mu]],coord[[Sigma]]]+Sum[affine[[Rho,Lambda,Mu]] affine[[Nu,Sigma,Lambda]]-affine[[Sigma,Lambda,Mu]] affine[[Nu,Rho,Lambda]],{Lambda,1,n}],{Mu,1,n},{Nu,1,n},{Rho,1,n},{Sigma,1,n}]

Defining the Riemann tensor with lower indices.

riemannDn:=riemannDn=Table[Simplify[Sum[metric[[Mu,Kappa]] riemann[[Kappa,Nu,Rho,Sigma]],{Kappa,1,n}]],{Mu,1,n},{Nu,1,n},{Rho,1,n},{Sigma,1,n}]

listRiemann:=Table[If[UnsameQ[riemannDn[[Mu,Nu,Rho,Sigma]],0],{Style[Subscript[R,Row[{coord[[Mu]],coord[[Nu]],coord[[Rho]],coord[[Sigma]]}]],16],"=",riemannDn[[Mu,Nu,Rho,Sigma]]}],{Nu,1,n},{Mu,1,Nu},{Sigma,1,n},{Rho,1,Sigma}]

Simplify[Simplify[TableForm[Partition[DeleteCases[Flatten[listRiemann],Null],3],TableSpacing->{2,2}]/.rule1]/.rule1]

Defining Ricci tensor:

ricci:=ricci=Table[Simplify[Sum[riemann[[Rho,Mu,Rho,Nu]],{Rho,1,n}]],{Mu,1,n},{Nu,1,n}]

listRicci:=Table[If[UnsameQ[ricci[[Mu,Nu]],0],{Style[Subscript[R,Row[{coord[[Mu]],coord[[Nu]]}]],16],"=",Style[ricci[[Mu,Nu]],16]}],{Nu,1,4},{Mu,1,Nu}]

TableForm[Partition[DeleteCases[Flatten[listRicci],Null],3],TableSpacing->{1,2}]

Defining Ricci scalar:

ricciscalar:=ricciscalar=Simplify[Sum[Sum[inversemetric[[Mu,Nu]] ricci[[Nu,Mu]],{Mu,1,n}],{Nu,1,n}]]

Simplify[Simplify[ricciscalar]]

rule2 = {(Sqrt[(x^2)+(y^2)+(z^2)])-> R};

rule3 = {((x^2)+(y^2)+(z^2))^(-1/2)-> R^(-1)};

rule4 = {((((x^2)+(y^2)+(z^2)))^(3/2))-> R^3};

rule5 = {((x^2)+(y^2)+(z^2))-> R^2};

rule6 = {Sqrt[R^2]^(-1)->R^(-1)};

RR=Simplify[Simplify[Simplify[Simplify[Simplify[Simplify[Simplify[ricciscalar]/.rule1]/.rule2]/.rule3]/.rule4]/.rule5]/.rule6]

Rtt:=(-B'[(Sqrt[(x^2)+(y^2)+(z^2)])] C'[(Sqrt[(x^2)+(y^2)+(z^2)])]+C[(Sqrt[(x^2)+(y^2)+(z^2)])] (((4/(Sqrt[(x^2)+(y^2)+(z^2)]))+((2 A'[(Sqrt[(x^2)+(y^2)+(z^2)])])/(A[(Sqrt[(x^2)+(y^2)+(z^2)])]))) B'[(Sqrt[(x^2)+(y^2)+(z^2)])]-(((B'[(Sqrt[(x^2)+(y^2)+(z^2)])])^2)/(B[(Sqrt[(x^2)+(y^2)+(z^2)])]))+2 B''[(Sqrt[(x^2)+(y^2)+(z^2)])]))/(4 ((C[(Sqrt[(x^2)+(y^2)+(z^2)])])^2)((B[(Sqrt[(x^2)+(y^2)+(z^2)])])^2))

LeftTerm:=Simplify[Simplify[Simplify[Simplify[Simplify[Simplify[Rtt/.rule1]/.rule2]/.rule3]/.rule4]/.rule5]/.rule6];

RightTerm:=Simplify[Simplify[Simplify[Simplify[Simplify[Simplify[(1/2)*RR*(-1/B[R])/.rule1]/.rule2]/.rule3]/.rule4]/.rule5]/.rule6];

LeftTerm - RightTerm

rule7={(2 ((C[R])^2)+(R^2) A'[R] C'[R]-2 R C[R] (3 A'[R]+R A''[R]))->STUFF1};

Simplify[LeftTerm-RightTerm/.rule7]

rule8={(R^2) C[R] ((A'[R])^2)->STUFF2};

Simplify[Simplify[LeftTerm-RightTerm/.rule7]/.rule8]

rule9={-4 ((A[R])^2) (C[R]-R C'[R])->STUFF3};

Simplify[Simplify[Simplify[LeftTerm-RightTerm/.rule7]/.rule8]/.rule9]

rule10={2 STUFF1 A[R]->STUFF4};

Simplify[Simplify[Simplify[Simplify[LeftTerm-RightTerm/.rule7]/.rule8]/.rule9]/.rule10]

rule11={STUFF2+STUFF3+STUFF4->STUFF5};

Simplify[Simplify[Simplify[Simplify[Simplify[LeftTerm-RightTerm/.rule7]/.rule8]/.rule9]/.rule10]/.rule11]

((-B'[R] C'[R]+C[R] (((4/R)+((2 A'[R])/(A[R]))) B'[R]-(((B'[R])^2)/(B[R]))+2 B''[R]))/(4 ((B[R])^2) ((C[R])^2)))+(1/(4 (R^2) ((A[R])^2) ((B[R])^3) ((C[R])^2)))(STUFF5 ((B[R])^2)+(R^2) ((A[R])^2) C[R] ((B'[R])^2)+R A[R] B[R] (-2 R C[R] A'[R] B'[R]+A[R] (R B'[R] C'[R]-2 C[R] (2 B'[R]+R B''[R]))))//ExpandAll

Simplify[Simplify[Simplify[Simplify[Simplify[LeftTerm-RightTerm/.rule7]/.rule8]/.rule9]/.rule10]/.rule11]//ExpandAll