f  =  
      Cos[ϕ]/
        Sqrt[r^2 + a^2 - 2 a r Sin[θ] Cos[ϕ]]

α  =  
      Integrate[f, {ϕ, 0, 2 π}, 
        Assumptions  ->  {a^2 + r^2 + 2 a r Sin[θ]  >  0, 
            a^2 + r^2 - 2 a r Sin[θ]  >  0 ,  a > 0,  
            r > 0,  θ  ∈  Reals}]

Expand[α]

na  =  Numerator[α]

da  =  Denominator[α]

Simplify[na]

Ac  =  Normal[Series[α, {r, Infinity, 6}]]

Ac[[1]]

Simplify[Ac[[4]]]

Simplify[Ac[[3]]]

Aϕ  =  
      Ac[[1]]  +  Simplify[Ac[[3]]] +  
        Simplify[Ac[[5]]]

t3  =  
      TrigReduce[LegendreP[3, 1, Cos[θ]]] /.  
        Sqrt[1 - Cos[θ]^2 ] ->  Sin[θ]

t5  =  
      TrigReduce [LegendreP[5, 1, Cos[θ]]] /.  
        Sqrt[1 - Cos[θ]^2] ->  Sin[θ]

t1  =  
      TrigReduce[LegendreP[1, 1, Cos[θ]]] /.  
        Sqrt[1 - Cos[θ]^2 ] ->  Sin[θ]

snew  =  
      Ac[[1]]/t1 (Subscript[P, 1, 1])[μ]   +  
        Simplify[Ac[[3]]/t3] (Subscript[P, 3, 1])[μ]  +  Simplify[
            Ac[[5]]/t5] (Subscript[P, 5, 1])[μ]

{t1, t3, t5}

Simplify[
      D[LegendreP[1, μ], μ]/LegendreP[1, 1, μ]]

LegendreP[3, 1, μ]

Simplify[
      Integrate[
          Cos[ϕ] LegendreP[3, Sin[θ] Cos[ϕ]], {ϕ, 0, 
            2 π}]/Sin[θ]]

TrigReduce[ LegendreP[3, 1, Cos[θ]]/Sin[θ]] /.  
      Sqrt[1 - Cos[θ]^2] ->  Sin[θ]