Integrate[(((x^(-7/3))Exp[-1/(x^2)])/((x+y))),{x,0,∞},Assumptions->{x>0,y>0,yc>0}]

i1=((θ^2)/π)Integrate[(((x^(-7/3))Exp[-1/(x^2)])/((((x+yc))^2)(x+y))),{x,0,∞},Assumptions->{x>0,y>0,yc>0}]

i2=FullSimplify[(i1/.{y->b(θc-θ),yc->b θc})+(i1/.{y->b(θc+θ),yc->b θc}),Assumptions->{b>0,θc>0,θ<θc,θ>-θc}]

iSing=-(1/(12 π))((((E^(-(1/((b^2) (((θ-θc))^2))))) ExpIntegralE[(7/6),-(1/((b^2) (((θ-θc))^2)))] Gamma[(1/6)])/(((θ-θc))^2))+((4 b (E^(-(1/((b^2) (((θ-θc))^2))))) ExpIntegralE[(5/3),-(1/((b^2) (((θ-θc))^2)))] Gamma[(2/3)])/(θ-θc)))

SeriesCoefficient[iSing,{θ,θc,1},Assumptions->{θc>0,θ<θc,b>0}]//N

t1=(1/π)Integrate[(((x^(-7/3))Exp[-1/(x^2)])/(x^(n+1))),{x,0,∞},Assumptions->{x>0,y>0,yc>0,n>0,n∈Integers}]

t1/.n->0//N

((4 b (E^(-(1/((b^2) (((θ-θc))^2))))) ExpIntegralE[(5/3),-(1/((b^2) (((θ-θc))^2)))] Gamma[(2/3)])/(θ-θc))+(-((4 b (E^(-(1/((b^2) (((θ+θc))^2))))) ExpIntegralE[(5/3),-(1/((b^2) (((θ+θc))^2)))])/(θ+θc))) Gamma[(2/3)]/.θ->0

i2/.θ->0

i3=Simplify[i2/.{b->1,θc->1}]

Plot[i3,{θ,-1.2,1.2},PlotRange->All,WorkingPrecision->20]

FullSimplify[i1]

i1/.y->