Solve integral $\int\frac{dx}{\sin x+ \cos x+\tan x +\cot x}$ I need to find:

$$\int\frac{1}{\sin x+ \cos x+\tan x +\cot x}\ dx$$

My attempts:
I have tried the conventional substitutions. I have tried the $\tan(x/2)$ substitutions, tried to solve it by quadratic but nothing has worked so far.

Partial Solution
$$\begin{aligned}
&\int\frac{\mathrm{d}x}{\sin(x)+\cos(x)+\tan(x)+\cot(x)}
\\&=\int\frac{sc}{s^2c+c^2s+1}\,\mathrm{d}x\quad *
\\&=\int sc\sum_{n\geq0}(-1)^n(s^2c+c^2s)^n\,\mathrm{d}x\quad\text{(Binomial series)}**
\\&=\int sc\sum_{n\geq0}(-1)^n\left(\sum_{0\leq k\leq n}\binom{n}{k}(s^2c)^{k}(c^2s)^{n-k}\right)\mathrm{d}x\quad\text{(Binomial theorem)}
\\&=\sum_{n\geq0}\,\sum_{0\leq k\leq n}(-1)^n\binom{n}{k}\int s^{n+k+1}c^{2n-k+1}\,\mathrm{d}x
\end{aligned}$$
Then the integral in the last expression can be computed iteratively, for $a,b\in\mathbb{N}$, as:
$$\int s^{a}c^{b}\,\mathrm{d}x
=-\frac{s^{a-1}c^{b+1}}{a+b}+\frac{a-1}{a+b}\int s^{a-2}c^{b}\,\mathrm{d}x$$
This solves the integral as a series but doesn't give a closed form and still has the caveat of infinite terms.

* $s=\sin(x),c=\cos(x)$
** The series is the expansion of $\big((s^2c+c^2s)+1\big)^{-1}$, which converges since $|s^2c+c^2s|<1$.
The iterative algorithm for $\int s^ac^b\,\mathrm{d}x$ is here, on Wikipedia.
