When hesitating to say something that might either hurt the other party or put the speaker in bad light, do the following expressions differ:




Or are they perfect synonyms?


いいづらい basically means you would feel kind of sore if you say it. You often use it when you feel it too harsh for the listener, or when the words are a tongue twister.

いいがたい is a relatively bookish word and you use it when the words are objectively not reasonable for the situation.

いいにくい is a general word that covers those two above.


In addition to what user4092 says,

You will say 楽{たの}しいとは言{い}いがたい when you are not pleasant or happy.
''~(し)がたい'' is often used when you don't want to do it or you can't do it because of the situation even if you want.
It is used with an intentional verb as ''許{ゆる}しがたい'', ''理解{りかい}しがたい'', ''耐{た}えがたい'', etc. It is not used when it's physically impossible, or with a non-intentional verb, so you can't say ''(!)霧{きり}で前{まえ}が見{み}がたい'', ''(!)sixthmonths は私には発音しがたい'', or ''(!)彼{かれ}の説明{せつめい}はわかりがたい''. (わかる is not an intentional verb, it is the intransitive form of わける, so わかりがたい can't be used, either わかりづらい or わかりにくい should be used.)
Furthermore, ''~がたい'' refers to yourself, not to others in most cases.

言いにくい is, as user4092 says, a general and neutral word, but has different nuance from 言いがたい. 言いがたい has a shade of meaning that you don't want to say, while, 言いにくい doesn't. So it is used when you say natural characteristics, your mind, features of electrical products, or the situation you face. It is not only used in written sentences but also conversation.

言いづらい is the most colloquial expression in these three words.
It is also used when you try to say a tongue twister in addition to what 4092 says. 
''~(し)づらい'' used with a transitive verb gives a hint that you actually want to do it and you can do it if you get over great difficulty.

If it is used with an intransitive, it means the same as '' ~(し)にくい''.

I give some examples as follows.

case 1. He told me he swam across the Pacific. I can't trust what he said.  
彼{かれ}の言うことは信{しん}じがたい。 It's appropriate.
! 彼の言うことは信じづらい。It's not appropriate. You'll wonder why I desire to trust him.
! 彼の言うことは信じにくい。It's not appropriate, too.  

case2. I want to ask his advice again, but I'm afraid he will get disgusted with me.
彼{かれ}に相談{そうだん}しづらい。 It's the most appropriate
彼に相談しにくい。 It's O.K. too, but it may be too objective.
! 彼に相談しがたい。 It's not used because I want to ask.
        If I don't want to ask him, it can be used, however 相談できない is more natural.

case3. The seats in airplanes does not burn easily. You explain it to someone.
飛行機{ひこうき}の座席{ざせき}は燃{も}えにくい。 It's appropriate; an objective fact
飛行機の座席は燃えづらい。 It's appropriate, the same as 燃えにくい, although a little colloquial.
! 飛行機の座席は燃えがたい。It's not used, quite strange.

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