Why does fundamental theorem of calculus not work for this integral $\int_0^{2\pi}\frac1{(3+\cos x)(2+\cos x)}$?

Because $$\frac{2\arctan(\frac{\tan(\frac x2)}{\sqrt3})}{\sqrt3} - \frac{\arctan(\frac{\tan(\frac x2)}{\sqrt2})}{\sqrt2}\tag1$$is not an antiderivative of $\frac1{(3+\cos x)(2+\cos x)}$. Such an antiderivative would have to be defined at every point of $[0,2\pi]$, but $(1)$ is undefined at $\pi$.

The anti-derivative $F(x)$ should be a Differentiable function over $(a,b)$.

Before using FTC, use Even symmetry Property which says that:$$\int_{0}^{2a}f(x)dx=2\int_{0}^{a}f(x)dx$$ When $f(2a-x)=f(x)$