How to get Texts from iphone to computer?

I have a iphone 4s and i want to upgrade to and iphone 5. I have a long list of very important texts. It would take days to screenshot and download the picture. Is there anyway I can get the texts downloaded on my PC?

Did you once come across the problem that your iPhone rejects receiving new SMS but all the SMS are so important that you don't want to delete them? Or you simply need to backup your iPhone SMS onto computer or print them or transfer to another iPhone?
Here I'd like to share an easy way on how to transfer your SMS from iPhone to computer and keep them in safe.
Things you need:
- iPhone
- computer
- Tansee iPhone Transfer SMS
The following are easy steps:
Step 1: Free Download Tansee iPhone Transfer SMS and install it.
Step 2: Connect iPhone to your computer.
Step 3: Launch Tansee iPhone Transfer SMS, the SMS contact info will be automatically displayed as shown in your iPhone screen
Step 4: If you want to copy all SMS to computer, click "Copy All SMS" on the top right of the first screenshot. And then select options:
1): Choose File Type: back up to TXT file or ANTS file; (The difference between TXT file and ANTS file lies in that ANTS file allows you to set a password for protection.
2): Choose File Path: set the destination path for the exported file.
3): Advanced Option: if you choose ANTS file format, you can add a password to protect the file.
Click OK Button to finish the Copy.