Question

Many solutions integral ?

Answer 1

In that case the multiple solutions would all be equivalent, differing by a constant is not hard to deal with typically: you just plug in a number for x in both answers that make it easy to compute (x=0, or 1, for certain answers can be easy at times) with a +C on one side and solve for C.

Answer 2

For real numbers the bottom two aren’t the same function. The middle is defined for 0 < x <= 1 and the bottom is defined for x >= 1. They’re derivatives would also be defined on their respective domains as well. The constant of integration is only because there’s an infinite number of antiderivatives since the derivative of a constant is zero.

Answer 3

Each connected component of the domain may have its own constant of integration.

Answer 4

>it gives the same thing if you derive them

They don't.  The derivative of the arctan expression is 1/(x\*sqrt(x-1)),  not 1/x\*sqrt(1-x))