All integrable functions have an infinite number of antiderivatives. This is why you have to include the integration constant after performing any indefinite integration.
Can a function have more than one different antiderivative? I keep getting different answers on some exercises and it drives me crazy, i double-checked my solving method and it's correct, yet i'm getting a different result.
Your answer is quite wrong. Unfortunately, you have shown no working at all, so the errors you are making cannot be pointed out.
Read this: integrate x/(x^4 - 2x^2 - 1) - Wolfram|Alpha (be sure to click on Show steps).