Here is one thing you can try.
(This is assuming two things. 1) that the integrated amp has a normal power switch and not a soft start switch and 2) does not have a 12v trigger in/on)
- Start with a good power strip or power conditioner that has a 12v on/off trigger built in.
- Plug the integrated amp into the strip/conditioner above
- Plug a 12v wall wart/power supply into the smart plug and the leads to the 12v trigger in. It can be a cheap one as you only need voltage to engage the trigger.
- Plug the smart plug into the wall.
- Then you can use the smart plug to trigger the integrated on/off
You can also do this with with some power conditioners that allow some of the outlets to be controlled independly and plug the smart plug into one outlet on the conditioner, which is always on, and run a cord from there to the trigger for another outlet. I have used an older Monster Power home theater conditioner this way for non-critical equipment to turn equipment on and off with voice commands. You can find the Monster Power's on eBay for >$100.
-Jeff