Before you do anything, you must figure out where it goes. It really has to vent outside. If it vents into the attic, you are going to have serious issues, including the potential of mold and rot as the hot, moist air is vented into an unconditioned space. It is (usually) against building codes to vent into the attic for this reason.
If it is venting through a roof or soffit vent, or out a wall, you're probably okay. It's best to keep runs as short as possible and with as few bends as possible. Dryer lint building up in the pipe can pose a fire hazard, so the shorter and straighter it is, the less chance there is to have anything build up, and the easier it is to clean out if needed.
Assuming it's vented properly, then I think boxing it in with drywall is a great way to go, and in fact is exactly what I did (in the corner):
You should also consider an in-line lint trap (I probably would have put one in if I knew about them):
I'd also recommend using straight vent pipe where possible, as it is smooth inside which reduces the opportunity for lint to attach and block the pipe. The rigid stuff you have is alright (way better than the white plastic crap) but it is still ridged inside.
Best Answer
One of the leading causes of house fires is dirty or plugged clothes dryer vents. Dryer vents should not be installed in ways that make them difficult to clean and inspect. Any vent in a wall cavity should be made of solid metal, with all joints sealed with metallic tape. There should not be any tight turns and they also need to be sized properly to the distance run. Flexible or corrugated plastic ducts should never be used in enclosed areas. When possible, it is safer to have vents exposed, as short as possible, made with metal duct with a minimum of bends. Ducts should be inspected for excess lint and vermin debris regularly and cleaned on regular intervals, usually every 6 months or annually depending on your usage. Also, anytime you see that your dryer is taking longer than usual to complete a drying cycle, it is a good idea to check the venting for problems.