"にくい(難い)" and "がたい(難い)" are an antonym of "やすい(易い)". "づらい(辛い)" is a synonym for "にくい(難い)".
"にくい(難い)" is in spoken language. It is used when we want to say something that is physically/technically difficult to do.
"づらい(辛い)" is in spoken language. It is used to say something that is mentally difficult to do. It indicates that you feel stressed out when you do it.
"がたい(難い)" is in written language. It is used when you want to say something that is extremely difficult/impossible to do. It is used for certain verbs that are related to understanding of someone's remarks/behaviors such as "理解しがたい", "想像しがたい", "信じがたい" and "認めがたい".
These three expressions are interchangeable depending on the context. I suggest that you memorize "にくい(難い)" and "づらい(辛い)".
Ken eats a lot of food but he is thin. He doesn't seem to put on weight.
- [OK]ケンは沢山食べるが、細い。太りにくいようだ。
- [NG]ケンは沢山食べるが、細い。太りづらいようだ。
- [NG]ケンは沢山食べるが、細い。太りがたいようだ。
It's hard for me to contact Ken because we had an argument and we haven't been keeping in touch for more than three months.
- [NG]ケンと喧嘩して、三ヵ月以上連絡していないから、連絡しにくい。
- [OK]ケンと喧嘩して、三ヵ月以上連絡していないから、連絡しづらい。
- [NG]ケンと喧嘩して、三ヵ月以上連絡していないから、連絡しがたい。
Ken often lies so it's difficult to believe what he says.
- [NG]ケンはよく嘘をつくので、彼の言う事は信じにくい。
- [OK(If you feel stress with Ken's remarks)]ケンはよく嘘をつくので、彼の言う事は信じづらい。
- [OK]ケンはよく嘘をつくので、彼の言う事は信じがたい。