I recently learned this nice shortcut from another SU user.
Select all cells as you usually do and then press
- Alt+Shift+, » Select only visible cells within your selection
- Ctrl+C » Copy
- Ctrl+V » Paste
- Alt+; » Select only visible cells within your (or Alt+Shift+, depending on keyboard layout)
- Ctrl+C » Copy
- Ctrl+V » Paste