Totally agree with everyone else, you have not failed.
I had similar with dd1 (18). My mum and SIL had straughtforward spontaneous vaginal births so my birth plan was the same however she was induced at 42wks, first with the pessary, then with drip. I don't think it's possible to breathe through drip based contractions there's no break between them! I had an epidural. She was back to back, ventoused out and I was left with a 3rd degree tear and episiotomy. I felt a total failure too, then I struggled to breastfeed, another 'natural' thing I didn't manage.
9years later I had dd2, my birth plan was give me a csection rather than stick me on the drip again. However fortunately her birth was spontaneous, so there was a gradual build up. I still had an epidural though.
Dd3 my birth plan was I like epidurals i want one. She was induced at 42wks but because by that point it was a path well travelled (by big babies) they induced by breaking my waters and she was born quickly, but obviously not painlessly!!. I wanted an epidural but there were no anaethetist! It was a good experience though, the polar opposite of my first.
I think the problem with hypnobirthing is people can attribute 'good' births to the hypnobirthing whereas perhaps they're just lucky. Much like when your baby is a good sleeper and you think well done me! Then you have another (2 in my case) who won't sleep.
Anyway talk to someone if you need to and congrats on your baby. And after all my twittering I am expecting number 4 in August and I did sign up to PBC early on as I'm still hoping for a breathing calm painless birth 