Bug introduced in 9.0 and fixed in 10.1
I want to integrate
$$\int_{0}^{2\pi} dt_1 \frac{a + b\cos(t_1 - t_2)}{c + d\cos(t_1 - t_2)}$$
where $a, b, c, d, t_2$ are real numbers and $c + d > 0$ & $0 \leq t_2 \leq 2\pi$.
I used a straightforward command in Mathematica (9.0.1.0),
Integrate[(a + b*Cos[t1 - t2])/(c + d*Cos[t1 - t2]), {t1, 0, 2 π},
Assumptions -> {a ∈ Reals, b ∈ Reals, c >= 0, d <= 0, c + d >= 0, 0 <= t2 <= 2π}]
to get an obviously incorrect answer: $$2\pi \frac{b}{d},$$ because clearly the integral should be $\propto c^{-1}$ for $|c| \gg |d|$.
My question is what am I doing wrong here? Is there a general lesson to be learnt here about how Mathematica handles this kind of integrations and the correct way to ask it to perform such integrations?