The difference between "習う", "学ぶ" and "勉強する"

Question: What is the difference between "習う", "学ぶ" and "勉強する"?

Some people answer this question such as:

学ぶ is Learn.
勉強 is Study.

But be careful, this answer is not correct.

At first, we should confirm the difference of "learn" and "study".

In English, 

"learn" means "to gain knowledge or skill of a subject", 

"study" means "to spend time reading, going to classes etc in order to learn about a subject".

So the following sentence is logically correct.

"I have studied Mathematics for many years, but I couldn't learn it"

"I learned something" means "I could get knowledge or skill of something".

"I studied something" means "I had spent time to learn something". We can't tell clearly from the sentence whether or not "He could get knowledge or skill of something".

In Japanese, "学ぶ" and "勉強する" are very similar. There is no difference like that of "learn" and "study".

The difference between "学ぶ" and "勉強する" is on the aspect of learning method.

For example, if you learned something undocumented know-hows through your experience, you can use the word "学ぶ" such as:

 "これらのノウハウは私が自分の経験を通じて学んだものです". 

In this case, most Japanese feel "学ぶ" is more suitable than "勉強する".

In contrast, if you learned something from books, you can use the word "勉強する" such as:

"私はこの本で数学を勉強しました". 

In this case, you can use "学ぶ" too.

But if you learned from a person, "勉強する" is NG, "学ぶ" is OK. Example as below.

I learned mathematics from Mr. Harada.

"私は原田先生から数学を学びました" is OK.

"私は原田先生から数学を勉強しました" is NG.

So, "学ぶ" can be used more widely than "勉強する".

Original meaning of the word "習う" is "imitate someone".

So "習う" requires some example.

So if you learned something undocumented know-hows through your own experience, you can't use the word "習う".

"学ぶ" has no such meaning of imitating. "習う" is used mainly for childrens' (under 10 years old) learning activity.