Can a continuous function have an indefinite integral that is discontinuous at a point?

Question

I was asked to evaluate the indefinite integral $\int \frac{dx}{2 + \sin x}$. I got the result as $$ \int \frac{dx}{2 + \sin x } = \frac{2}{\sqrt{3}} \arctan \bigg( \frac{2\tan(\frac{x}{2}) + 1}{\sqrt{3}} \bigg) + C $$ $f(x) = 2 + \sin x$ is continuous for all $x \in \mathbb{R}$, though its anti-derivative is discontinuous at $x = (2k + 1) \pi$ and thus non differentiable at $x = (2k + 1)\pi$ for $k \in \mathbb{Z}$. Does this mean the indefinite integral is invalid at $x = \pi$ and its multiples?