Complex and complicated are often used as synonyms but they are different. Intricate is very similar to complicated, but expresses the amount off detail more than how complicated it is. If something is complex it means it has many different components. However it doesn't necessarily mean that it's difficult to understand. If something is complicated it doesn't necessarily mean it has many components but it will be hard to understand. If something is intricate it means it is very detailed and possibly complicated.

Something that is complicated usually has a lot of details or factors or moving parts. Something complicated has a lot of things to keep track of and remember. You could say Physics is complicated, or driving is complicated. Intricate tends to be used more with physical objects, to refer to the amount of details, for instance a design or a picture is intricate if it contains a lot of lines or shapes. Intricate can also be used to refer to things like a plan, or anything someone makes or comes up with that is very detailed. For example “an intricate plot to cheat on the exam” Complex tends to be used for more abstract things, for instance you can say “the history of the US is complex” because there’s a lot of concepts and opinions involved. Complex can also imply there’s a lot of ambiguity depending on the context. But complex can also be used to refer to physical things, like “the highway system is very complex”. Complicated and complex are very similar and for the most part can be used interchangeably, this is just my interpretation as a native speaker. The only time I would be careful is if you are describing a person. If you call someone complicated it is more of an insult, insinuating they might be difficult or high maintenance. If you call someone complex, it is more of a compliment, insinuating they are intellectual or deep, or possibly mysterious. Hope this helps!

They're synonyms so there's no point trying to differentiate them :)