Both are natural and common.
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Persian LearnerIn all contexts?I'm not sure what you mean. The sentences are not slang or otherwise limited in usage.
AlpheccaStarsI'm not sure what you mean. The sentences are not slang or otherwise limited in usage.You know, a few days ago I had a grammar exam, and in the gap filling section I used a to-infinitive to fill the gap after bother.
Persian LearnerI can't remember the exact sentence. I want to know whether it is possible to use either a to-infinitive or a gerund after bother (as a verb) in all situations/sentences without any difference in meaning?Yes.
Persian Learnercorrect and natural?Yes. 'bother to do' is used more than 'bother doing', but both are correct.
Either the infinitive or the gerundial form is fine. There may be some contexts in which one or the other is more idiomatic, but in most contexts each is as good as the other.