BOOK

I've been writing the a book about English grammar in Japanese by with iBook Author of for Mac.

"The" implies that you are writing the only book about English grammar in Japanese. "A" means that you are writing one book, but others exist.

The reason why I stared do it I decided to put contents I've learned in the time of examination for university in order.

I think I would probably say, "I decided to do this because I wanted to put all of the things I've learned for university exams in one place and in order.

I was able to know that writing a book is so difficult.

I would say:
"I now know that writing a book is very difficult."
"I found out that writing a book is very difficult."
"Writing a book is very difficult."