What is the rule on using a "zu" phrase with a separable verb? Between
Ich fange an zu weinen.
Ich fange zu weinen an.
Are both correct? If the "zu" phrase is long, would that mean the second variant is less appealing?
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Both constructions are correct.
The first example is a coherent construction, where the (extended) infinitive group is part of the main clause. (ext. inf. group = additional words + zu + infinitive)
The second example is an incoherent construction, where the (extended) infinitive group is appended. This construction sometimes must be separated with a comma.
Ich fange (bitterlich) zu weinen an.
This is a coherent construction. Think of "zu weinen" as an object instead, like "einen Aufsatz" in
Ich fange einen Aufsatz an.
Ich fange an, (bitterlich) zu weinen.
Incoherent construction. What you are starting is specified by the appended (extended) infintive group. A comma seems optional, although I would recommend one for extended constructions.
The rule is called "Infinitivsatz" or "Infinitivkontruktion". Further informations about the construction (in German!): https://de.wikipedia.org/wiki/Nebensatz#Infinitivs.C3.A4tze
A "dass" usually avoids the usage of Infinitivsätzen. Therefore, the verb has to be inflected:
Ich freue mich, dich zu treffen.
Ich freue mich, dass ich dich treffe.
They're both correct and express the same.
In my opinion, neither sounds less appealing. An example:
Ich freue mich, dich nach dieser langen anstrengenden ermüdenden Zeit zu sehen.
Ich freue mich, dich zu sehen nach dieser langen anstrengenden ermüdenden Zeit.
If I'd have to choose, I'd take the first one, but in my opinion, they're both alright. The same is true for your two sentences.
Hopefully, that was what you were asking for.