i need to dig out an old very long privit what is the best tool i can buy or hire to do it

just been looking through the HSS hire guide, the only thing I could see is the mini skidsteer, and the tool with it the ripper,and the carryall that goes with it would no dout save your back,it looks as if it would make the job a dodle, cost for 24hrs which should be more than enough without any sweet,£106.50 ripper to go with it £8.50 carryall £6.50, £121.50,but you get the job dun and have a fun day with no sweet or backake,

Apologies for this is not an answer to your question: Your question brought back memories that made me really smile. When I had my first home aged 21 I was fired up to tackle everything on my own. Although I was a healthy and fit young lady (all those years ago LOL) I wasn't strong enough to dig out the roots. I worked out a method of tying a rope around each bush and secured that to the tow bar on the back of my car. The neighbours must have thought I was totally bonkers when I took off slowly up the road tearing each bush out. But hey, it worked!! It's only now I am a LOT LOT older that I realise how daft I must have looked. I wouldn't mind having that level of enthusiasm and determination now!!

Sprog's method can be the best. really. I have pulled out quite large trees by tying a rope on high up with the other end anchored to the base of something really secure, then winching it over onto its side when the roots can then be severed more easily.
Another solution which works: Use a brushwood killer and spray onto the leaves and around the root area. If you are not going to plant anything in its place, then you could try something I've done several times now with big tree roots, albeit unintentionally. That is to introduce honey fungus. You may not like the idea of doing this but it works very effectively; every space in the ground that was once occupied by a root becomes filled with rotted root remains that in a year or so has completely disappeared.