Closed form of the integral of $r \ln (r^2 + \rho^2 - 2r \rho \cos (\theta - t))$

Question

I have the following integral that needs to be evaluated.

$$ \int_{0}^{2\pi} \int_0^{\frac 1 {\sqrt{4 \cos^2 \theta + 9 \sin^2 \theta}}} \frac r 2 \ln (r^2 + \rho^2 - 2r \rho \cos (\theta - \phi)) dr d\theta $$

I tried using Mathematica but it gives the following output

-((\[Rho] (-2 E^(2 I \[Theta]) Sqrt[
        26 - 5 E^(-2 I \[Theta]) - 
         5 E^(2 I \[Theta])] + (5 - 26 E^(2 I \[Theta]) + 
          5 E^(4 I \[Theta])) \[Rho] ArcTan[
         Cot[\[Theta] - \[Phi]]] Cos[
         2 \[Theta] - 2 \[Phi]] Csc[\[Theta] - \[Phi]] - (5 - 
          26 E^(2 I \[Theta]) + 5 E^(4 I \[Theta])) \[Rho] ArcTan[
         Cot[\[Theta] - \[Phi]] - (
          Sqrt[2] Csc[\[Theta] - \[Phi]])/(\[Rho] Sqrt[
           13 - 5 Cos[2 \[Theta]]])] Cos[
         2 \[Theta] - 2 \[Phi]] Csc[\[Theta] - \[Phi]] - 
       10 \[Rho] Cos[\[Theta] - \[Phi]] Log[\[Rho]] + 
       52 E^(2 I \[Theta]) \[Rho] Cos[\[Theta] - \[Phi]] Log[\[Rho]] \
- 10 E^(4 I \[Theta]) \[Rho] Cos[\[Theta] - \[Phi]] Log[\[Rho]] + 
       5 \[Rho] Cos[\[Theta] - \[Phi]] Log[(
          5 \[Rho]^2 + 5 E^(4 I \[Theta]) \[Rho]^2 - 
           2 E^(2 I \[Theta]) (2 + 13 \[Rho]^2))/(
          5 - 26 E^(2 I \[Theta]) + 5 E^(4 I \[Theta])) - (
          4 \[Rho] Cos[\[Theta] - \[Phi]])/Sqrt[
          26 - 5 E^(-2 I \[Theta]) - 5 E^(2 I \[Theta])]] - 
       26 E^(2 I \[Theta]) \[Rho] Cos[\[Theta] - \[Phi]] Log[(
          5 \[Rho]^2 + 5 E^(4 I \[Theta]) \[Rho]^2 - 
           2 E^(2 I \[Theta]) (2 + 13 \[Rho]^2))/(
          5 - 26 E^(2 I \[Theta]) + 5 E^(4 I \[Theta])) - (
          4 \[Rho] Cos[\[Theta] - \[Phi]])/Sqrt[
          26 - 5 E^(-2 I \[Theta]) - 5 E^(2 I \[Theta])]] + 
       5 E^(4 I \[Theta]) \[Rho] Cos[\[Theta] - \[Phi]] Log[(
          5 \[Rho]^2 + 5 E^(4 I \[Theta]) \[Rho]^2 - 
           2 E^(2 I \[Theta]) (2 + 13 \[Rho]^2))/(
          5 - 26 E^(2 I \[Theta]) + 5 E^(4 I \[Theta])) - (
          4 \[Rho] Cos[\[Theta] - \[Phi]])/Sqrt[
          26 - 5 E^(-2 I \[Theta]) - 
           5 E^(2 I \[Theta])]]) Sin[\[Theta] - \[Phi]])/(5 - 
     26 E^(2 I \[Theta]) + 5 E^(4 I \[Theta])))

Does closed form of the above integral exists? What approaches might I try to solve this integral? Is this possible to express this integral in terms of special functions?