What is the indefinite sum of the tangent function, that is, the function $T$ for which

$\Delta_x T = T(x + 1) - T(x) = \tan(x)$

Of course, there are infinitely many answers, who all differ by a function of period 1. Ideally, I would like the solution to be of the form

$T(x) = $ nice_function$(x)$ + possibly_ugly_periodic_function$(x)$, where nice is at least piece-wise continuous.

If any of the following sums can be found, the sum of tan can also be found:

• $\sum \sec x$

• $\sum \csc x$

• $\sum \cot x$

• $\sum \frac{1}{e^{ix} + 1}$

I have tried several methods without success, including using a newton series (which does not converge for non-integer $x$), and trying to guess possible functions.

I would also appreciate lines of attack if a solution is not known.