You focus on these verb patterns at upper-intermediate level, and the grammar is called "causative verbs". There are only four: help, make, let and have. (Jerry, I suspect you're also thinking of stative verbs/verbs of sense in your list.)

I'm very surprised that someone led you to believe that "help me to do it" is the only acceptable way of writing the phrase. Both forms are possible, and removing "to" started happening about 150 years ago. The "help me do it" pattern has become more popular since around 1950.

Unfortunately, "help" is not a good example, as it is the only verb which can use or omit "to". All the other verbs follow clear patterns.

You can use "help me do it" in academic writing. There's absolutely nothing wrong with that.

Both “help someone do something” and “help someone to do something” are acceptable.<o:p></o:p>

The form without “to” seems to be more common in everyday speech than the form with “to” (especially in American English), but both forms are common in formal writing and “to do it” is slightly more formal:<o:p></o:p>

<o:p> </o:p>

He helped me move to London. (correct, more common)<o:p></o:p>

He helped me to move to London. (correct, less common in spoken language)<o:p></o:p>

<o:p> </o:p>

The lesson helped me understand the grammar. (correct, more common)<o:p></o:p>

The lesson helped me to understand the grammar. (correct, less common in spoken language)<o:p></o:p>

<o:p> </o:p>

My parents helped my brother realise/realize his mistake. (correct, more common)<o:p></o:p>

My parents helped my brother to realise/realize his mistake. . (correct, less common in spoken language)<o:p></o:p>

<o:p> 

</o:p>

… helped me do it (correct, more common)

<o:p></o:p>

… helped me to do it (correct, less common in spoken language)<o:p></o:p>

not native but i know the answer :D

both work ..... but  "to" is used less often .