Definite integral impossible but indefinite possible

Question

I am trying to do a spherical integral in Mathematica:

f[θ_, ϕ_] = (k^4 Cos[θ]^2 Cot[β] Csc[β] Sin[θ]^3 Sin[ϕ]^2)/(
  4 (1 + k Cos[θ]) (1 + k Cos[β] Cos[θ] + k Cos[ϕ] Sin[β] Sin[θ])) + 
  (k^4 Cos[θ] Cos[ϕ] Csc[β] Sin[θ]^4 Sin[ϕ]^2)/(
  4 (1 + k Cos[θ]) (1 + k Cos[β] Cos[θ] + k Cos[ϕ] Sin[β] Sin[θ]))

If I try the following:

Integrate[f[θ, ϕ], {ϕ, 0, 2 π}]

Mathematica does not produce an answer. However, if I try to do an analytical integral like this:

g[θ_] = Integrate[f[θ, ϕ], ϕ]

I get an answer very quickly, and that answer differentiates to the integrand I started with. Now I can evaluate the value of the integral I am trying to find by doing:

g[2 π] - g[0]

My question is twofold:

1. Why is this happening in Mathematica? I assume it is because the algorithms for finding a definite and indefinite integrals are different.
2. Is this a valid way of doing the integral in this case (and others)? I am asking because the end result does not agree with a purely numerical integral over the sphere, and this might be the source of error.

Answer 1

It seems like you just weren't patient enough. After 12.5 minutes, version 11.1.1 gives me an answer. Not a beautfiul one, but not a ridiculuous one, either. We've made this better in 11.2. It takes it a little over a minute to come up with a simpler answer.

Re 1): It might use different methods for definite and indefinite integration (e.g., residues aren't of much use for indefinite integrals). It could also be that it's checking that the antiderivative is is continuous over the interval of integration / trying to patch pieces together to find a continuous one. Without really going into the guts of Integrate, I couldn't tell you which.

Re 2): Given 1, that's dangerous. Not to claim that Integrate is perfect, but if it could do the integral by a simple application of the fundamental theorem, it probably would have, so you should check such answers carefully.