Define spherical harmonics (fully normalized).

P[n_] := Simplify[1  / ((2^n) Factorial[n]) D[((((t^(2 ))- 1))^n), {t,n}]]

Pa[n_,m_] := (((1 - (t^2)))^(m/2))D[P[n], {t,m}]

Shz[n_,ω_] :=FullSimplify[ ReplaceAll[1/Sqrt[4 π] * Sqrt[2n + 1] * P[n],t->Cos[ω[[1]] ]]]

Shc[n_,m_,ω_] := FullSimplify[1/Sqrt[4 π] Sqrt[( 2(2n + 1) Factorial[n-m]) / Factorial[n + m]] Cos[m ω[[2]] ] ReplaceAll[Pa[n,m],t->Cos[ω[[1]] ]]]

Shs[n_,m_,ω_] := FullSimplify[1/Sqrt[4 π] Sqrt[(2(2n + 1) Factorial[n-m]) / Factorial[n + m]] Sin[m ω[[2]] ] ReplaceAll[Pa[n,m],t->Cos[ω[[1]] ]]]

ω = {θ, ϕ};

Integral of terms resulting from analytic derivation (with x / y / z replace by the appropriate expressions in spherical coordiantes).

xz

Integrate[(Sin[θ]Cos[ϕ])*( Cos[θ]) * Sin[θ], {θ, 0, π/2}, {ϕ, 0, 2 π}]

zy

Integrate[(Sin[θ]Sin[ϕ])*( Cos[θ]) * Sin[θ], {θ, 0, π/2}, {ϕ, 0, 2 π}]

zz

Integrate[(Cos[θ])*( Cos[θ]) * Sin[θ], {θ, 0, π/2}, {ϕ, 0, 2 π}]

Project the kernel into spherical harmonics to show the equivalence to the results by Ramamoorthi and Hanrahan [2001] and Basri and Jacobs [2003].  See their papers for the numerical values.

N[Integrate[Shz[0,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]

N[Integrate[Shz[1,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]

N[Integrate[Shz[2,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]

N[Integrate[Shz[3,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]

N[Integrate[Shz[4,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]

N[Integrate[Shz[5,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]

N[Integrate[Shz[6,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]

Spherical harmonics terms with m \neq 0 are zero.

N[Integrate[Shc[1,1,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]

N[Integrate[Shs[1,1,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]

N[Integrate[Shc[2,1,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]

N[Integrate[Shc[2,2,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]

N[Integrate[Shs[2,1,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]

N[Integrate[Shs[2,2,ω]*(2π/3) Cos[θ] Sin[θ], {θ,0,π/2}, {ϕ,0,2π}]]