<s>[INST] What is the difference between $\ln(\sin\frac{x}{2}+\cos\frac{x}{2} )- \ln(\sin\frac{x}{2}-\cos\frac{x}{2})$ and $\ln(\tan x+\sec x)$ I've noticed if you ask wolfram what is the integral of $ \sec x$ they will answer 
$$\ln\left(\sin\frac{x}{2}+\cos\frac{x}{2}\right) - \ln\left(\sin\frac{x}{2}-\cos\frac{x}{2}\right)$$
I've been taught that the $\int\sec x = \ln|\tan x + \sec x | + C$ which is in my opinion way simpler to read and write.
However, even in the alternatives form of Wolfram, I could not find it there which let me believe that what I've been taught is probably not exact (but near) and they teach us that so it is simpler for us to learn.
Am I right ? If yes, what is the difference ?
 [/INST] there is no difference. here is a reason 
$$\begin{align}\tan x + \sec x &= \frac{\sin x}{\cos x} + \frac 1 {\cos x}\\
& = \frac{1+\sin x}{\cos x} \\
&= \frac{\sin^2x/2 + \cos^2x/2 + 2\sin x/2 \cos x/2}{\cos^2 x/2 - \sin^2 x/2} \\
&= \frac{(\sin x/2 + \cos x/2)^2}{(\cos x/2 - \sin x/2)(\cos x/2 + \sin x/2)}\\
&=\frac{\sin x/2 + \cos x/2}{\cos x/2 - \sin x/2}\end{align}$$
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