I know that similar functions can have very different antiderivatives. For example, even though int((x)/((x^4)+1)) dx looks very similar to int(((x^4)+1)/(x)), their solutions are very different. How can I prove that I'm right by solving both?

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you wanna indefinite integral both of them right?

Yes.

the last one just split the fraction:
x^4/x +1/x and integrate

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