Deflection Angle for Schwarzschild Metric

f=1-2*(m/r)
metric=DiagonalMatrix[{-f,(1/f),(r^2),(r^2)((Sin[θ])^2)}]

met=-f (dt^2)+(1/f)(dr^2)+(r^2)((dϕ)^2);

gtt=Coefficient[met,dt,2];
grr=Coefficient[met,dr,2];
gϕϕ=Coefficient[met,dϕ,2];

dt2=(1/(f^2))(dr^2)+(1/f)((dϕ)^2)

grr1=((((grr/(-gtt))))^(1/2))//FullSimplify

gff1=(((((gϕϕ)/(-gtt))))^(1/2))//FullSimplify

κ1=(1/grr)D[gff1,r]//FullSimplify//PowerExpand

r=(1/u);
A=-f;
B=((-1)/f);
c=-(r^2);

du=(c ((u^4)/B))((c/(A (b^2)))-1)//FullSimplify

d2u=(1/2)D[du,u]

FullSimplify[Series[d2u,{u,0,2}]//Normal//Expand]

u=((Sin[ϕ])/b)+F[ϕ];

D[D[u,ϕ],ϕ]//FullSimplify

DSolve[((-Sin[ϕ])/b)+F''[ϕ]+u-3m(((((Sin[ϕ])/b)))^2)==0,F[ϕ],ϕ]//FullSimplify

u=((Sin[ϕ])/b)+((m (3+Cos[2 ϕ]))/(2 (b^2)));

κ1//FullSimplify//PowerExpand

k1=((Sqrt[(1/u)](1-3m*u))/(Sqrt[-2m+(1/u)]))

k2=Normal[Series[k1,{b,∞,2}]]//FullSimplify

-((-4 (b^2)+15 (m^2)+(m^2) Cos[2 ϕ]+8 b m Sin[ϕ])/(4 (b^2)))//Expand

k3=((15 (m^2))/(4 (b^2)))+(((m^2) Cos[2 ϕ])/(4 (b^2)))+((2 m Sin[ϕ])/b)

Integrate[k3,{ϕ,0,Pi}]//FullSimplify