How to extract the historical tweets from twitter API?

We need the historical tweets for some movies. Right now, we tried the streaming API and search API from twitter. The streaming API could not give us a parameter to choose the time range we need and the search API could only give us data one or two weeks in advance. Is there a way for us to extract the historical tweets from 2014-05-01 to 2014-07-01 (For example)? I found the following ways that is possible to do that: 1: Twitter advanced search https://twitter.com/search-advanced?lang=en It could find the search result I need. But how could I download the search result? Is there anyway to write a code and save the search result?

2:Using the twitter analytic website like topsy. But it also has the difficulty to save the result.

3: It seems that some packages like twitter4J could help with that: http://ift.tt/MbOQ8l Is there any python or R package could help us to do that?

4: We need this data to do a research. It is not a good choice to spend a long time for the extraction of the data. Is there anyway to buy this data from some professional website?

Thanks a lot!