Incorrect result of integration with GenerateConditions

Question

The GenerateConditions->True option of the Integrate command is useful and works properly in most cases. However, there are cases when this command works improperly. Here is one of these: the result of

Integrate[1/(1 - Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True]

ConditionalExpression[Cot[a/2] - Cot[b/2], a^2 < a b || a b > b^2]

is not correct if the integration range includes a zero of Cos[x]-1.

The result of

Integrate[1/(1 - Cos[x]), {x, Pi/4, b}, Assumptions -> {b} \[Element] Reals, GenerateConditions -> True]

-Cot[b/2] + Cot[\[Pi]/8]

is not better. The same issue in 12.3.1 on Windows 10 with Integrate[1/(Sqrt[3] - 2*Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True] etc.

Is there a workaround?

Answer 1

The integral

Integrate[1/(1 - Cos[x]), {x, 0, 2 Pi}  ]
(*... does not converge on {0,2 \[Pi]}*)

does not converge.

That's why you have to restrict your assumptions for example to 0<a<b<2Pi !