<s>[INST] Integral $\int \frac{dx}{\tan x + \cot x + \csc x + \sec x}$ $$\int \frac{dx}{\tan x + \cot x + \csc x + \sec x}$$
$$\tan x + \cot x + \csc x + \sec x=\frac{\sin x + 1}{\cos x} +\frac{\cos x + 1}{\sin x} $$
$$= \frac{\sin x +\cos x +1}{\sin x \cos x}$$
$$t= \tan {\frac{x}{2}}$$
On solving ,
$$\frac{1}{\tan x + \cot x + \csc x + \sec x}=\frac{t(1- t)}{1+ t^2}$$
$$\implies \int \frac{\tan {\frac{x}{2}}(1-\tan {\frac{x}{2}})}{1+\tan^2 {\frac{x}{2}}}{dx}$$
I think, I have made the things more difficult. How can I proceed further? Is there any better substitution for it?
 [/INST] $$\begin{aligned}\int \frac{1}{\frac{\sin(x)}{\cos(x)}\:+\:\frac{\cos(x)}{\sin(x)}\:+\:\frac{1}{\sin(x)}\:+\:\frac{1}{\cos(x)}}dx
& = \int \:\frac{\sin \:\left(2x\right)}{2\left(\cos \:\left(x\right)+\sin \:\left(x\right)+1\right)}dx
\\& =\frac{1}{2}\cdot \int \:\frac{\sin \left(2x\right)}{\cos \left(x\right)+\sin \left(x\right)+1}dx
\\& =\frac{1}{2}\cdot \frac{1}{2}\cdot \int \:\frac{\sin \left(t\right)}{\sin \left(\frac{t}{2}\right)+\cos \left(\frac{t}{2}\right)+1}dt
\\& =\frac{1}{2}\cdot \frac{1}{2}\cdot \int \:\sin \left(\frac{t}{2}\right)+\cos \left(\frac{t}{2}\right)-1dt
\\& =\color{red}{\frac{1}{4}\left(-2x+2\sin \left(x\right)-2\cos \left(x\right)\right)+C}
\end{aligned}$$
Applyied substitution: $$\color{blue}{t=2x,\quad \:dt=2dx}$$
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