I tried to find an answer to this question but found nothing, maybe because I don't know how to properly formulate it. I hope I can make myself understood here.
The thing is, recently I started learning about indefinite integrals in school and my teacher told us that the derivatives of inverse trigonometric functions have two indefinite integrals. For example:
$$\int\frac1{1+x^2}dx=\arctan(x)+C=-\text{arccot}(x)+C$$
Which makes sense since the derivatives of related inverse trigonometric functions differ just in the sign. Nonetheless, this clashes with my notion that there must be only one result for any given argument of a function. Moreover, when I looked it up on the Internet I noticed that most websites just show the positive indefinite integrals. My question is whether the indefinite integrals of the derivatives of inverse trigonometric functions have in fact two different results and, if that's the case, why is it like that?