abuse of using conventional and nonconventiaonl arcsec range

From a very quick chuck it into wolfram
https://www.wolframalpha.com/input/?i=integrate+1%2Fsqrt%28x%5E2+-+2%29
The integral is
arctanh(x/sqrt(x^2-2))
which is real in -sqrt(2)<x<sqrt(2). Outside that interval, the integral is complex.

Not sure what you're meaning here - certainly x/sqrt(x^2-2) is complex (not real) if -sqrt(2)<x<sqrt(2).

It seemed like the OP was treating things as real outside that interval.

Surely it *is* real outside the interval (-sqrt{2}, sqrt{2})?

I want to hear some justification from people who take conventional arcsec x range and also conveniently use sqrt(x^2 - 1) = tan t for all possible x values. Their answer is correct indeed, because the negative sign is cancelled in the middle, miraculously for them, but they never anticipated it. when x<=-1 and the moment they wrote int sec t dt instead of - sec t, it is false. Isn't this a kind of abuse, like abuse of notation? Please note that this problem does not necessarily apply on the case where hyperbolic trigonometric identities and substitutions are used. I'm only curious about x = sec t substitution case.

I'm not sure what you're actually asking here. I am fairly confident the number of people on TSR who've read through your thread, have the inclination/ability to be able to meaningfully answer this, *and* assume $\sqrt{\sec^2 t - 1} = \tan t$ for all values of t is exactly zero.

To be slightly more helpful here:

When faced with a problem where a function is defined over disconnected intervals, experienced mathematicians tend to be fairly wary of "universal solutions" that try to work over all the intervals at once. At university level it's pretty much unheard of to say $\int \frac{1}{t} \,dt = \log |x|$ rather than log x, for example. In a problem like this their natural impulse will be to transform the "ugly" domain into a more familiar one (in this case using the obvious fact that for $b > a > \sqrt{2}$, $\int_{-b}^{-a} \sqrt{x^2-1}\,dx = \int_a^b \sqrt{x^2-1}\,dx$).

Not only does this mean that they wouldn't ever hit the issues you've been having, it also means they're not very experienced, or, bluntly, interested, in tracking down exactly where an ambiguous square root sign / choice of domain / reversal of integral limits / misuse of modulus occurs in a problem like this. If it was a few lines of working, sure, but multiple pages and it's like "I'm not getting paid for this you know!".

Of course it's obvious when f(x) = f(-x), or a function is an even function, then the endpoints of the interval can also be reflected in the y-axis, and it gives the same value for the integral or area. You have misunderstood the original post somehow, also this fact was used to prove that S_ii = - ln | sec t + tan t | which contradicts the universal solution S you get in terms of parameter t through the "usual" method. I hope that's the only thing that you misunderstood.

I don't understand what you meant by "assume for all values of t is exactly zero." If you meant "assume tan t = positive square root of (sec^2t -1) for all values of t", according to the OP, how can a positive value be equal to negative value? If you meant "assume t is exactly zero", then I understand your suspicion that I'm getting paid for posts on TSR.