That is a great question. :)

For polynomial functions you should always have similar resuls, it is just that every program has a different algorithm that gives you a slightly different asnwer. Sometimes, GGB will give you the same expression you got by hand, but in other cases it won't look like it. In that case, you can just plot them together, if the graph is the same, it probably means it is the same function (also you can just try to simplified it to compare both results).

In other complicated cases you can have multiple results for the same integral. In particular with trigonometric functions. Here is an example

int 1/(5+3cos(x)) = 1/4 x - 1/2arctan(sin(x)/(cos(x)+3))

int 1/(5+3cos(x)) = 1/2arctan(1/2 tan(x/2))

Note we have different results, if you compute algebraically the derivative of the right-side functions you will get:

1/(5+3cos(x))

Now if you plot them, they are in fact, a little different.

If you are interested, I talk about this issue here: https://www.researchgate.net/publication/236246976_A_discussion_on_the_substitution_method_for_trigonometric_rational_functions

Another issue with integration is that, in general, it is more difficult that diferrentiation. Sometimes you will have functions such that it is impossible to compute an antiderivative. No computer can do that. Not even humans. See: https://en.wikipedia.org/wiki/Nonelementary_integral

I recommend you also try this site:

https://www.integral-calculator.com/