tl;dr The results you're seeing are correct. You're seeing a branch cut

The logic you've used here is really common even among very smart engineers and proffesors, but not correct. They don't have to be equal.

Lets assume you have two expressions called "a" and "b". You know "a" and "b" are equal. Their indefinite integrals have to be equal too right? Integral a(x) d(x) has to be equal to Integral b(x) d(x)? No, they don't have to be equal. It is very common for them to be equal, but there are three major reasons why you will not see them being equal:

First, the values of the indefinite integrals can always differ by a constant. Sometimes you the constant difference won't be obvious in the resulting formula like it would be in most calculus examples of this.

Second, where you are using inverse functions such as Log, Sqrt, ArcSine, etc, there are branch cuts. These will cause the results to differ by a piecewise constant function. This is what is happening in your example. If you subtract the two results you get and Simplify under the assumption that their input is a real number, you get a piecewise constant function equal to PiI for x less than zero and equal to -PiI for x greater than zero

Lastly, integration is often done "generically", meaning we basically ignore the value at some isolated points. A good example of this is that we say the integral of x^n is x^(n+1)/(n+1). This isn't true for n=-1 in which case the integral is Log[x].

In[2]:= Integrate[-1/(x - 1), x]

Out[2]= -Log[1 - x]

In[3]:= Integrate[1/(1 - x), x]

Out[3]= -Log[1 - x]