Rubi can't integrate this, but Mathematica can

[darthvlado]

In[1]:= $Assumptions={0<v<1};
In[2]:= Integrate[1/(2Pi) Sin[\[Theta]]/2 (v^2 (Sin[\[Theta]]Cos[\[Phi]])^2)/Sqrt[1-v^2 (Sin[\[Theta]]Cos[\[Phi]])^2],{\[Phi],0,2\[Pi]},{\[Theta],0,\[Pi]}]
Out[2]= (-v Sqrt[1-v^2]+ArcSin[v])/(2 v)
In[3]:= <<Rubi`
In[17]:= Int[Sin[\[Theta]]/2 (v^2 (Sin[\[Theta]]Cos[\[Phi]])^2)/Sqrt[1-v^2 (Sin[\[Theta]]Cos[\[Phi]])^2],{\[Theta],0,\[Pi]}]
Int[1/(2Pi) %,{\[Phi],0,2\[Pi]}]
Out[17]= -(1/2)+1/2 ArcTanh[v Cos[\[Phi]]] (v Cos[\[Phi]]+Sec[\[Phi]]/v)
Out[18]= -Underscript[\[Limit], \[Phi]->0]((Sqrt[1-v^2] ArcTan[Sqrt[1-v^2] Cot[\[Phi]]])/(4 \[Pi])+(v ArcTanh[v Cos[\[Phi]]] Sin[\[Phi]])/(4 \[Pi])+Int[ArcTanh[v Cos[\[Phi]]] Sec[\[Phi]],\[Phi]]/(4 \[Pi] v))+Underscript[\[Limit], \[Phi]->2 \[Pi]]((Sqrt[1-v^2] ArcTan[Sqrt[1-v^2] Cot[\[Phi]]])/(4 \[Pi])+(v ArcTanh[v Cos[\[Phi]]] Sin[\[Phi]])/(4 \[Pi])+Int[ArcTanh[v Cos[\[Phi]]] Sec[\[Phi]],\[Phi]]/(4 \[Pi] v))

[stblake]

Rubi really shouldn't be used for definite integration. I would be in favour of removing the definite integration feature/rule in Rubi.