Is it normal for my partner to not discipline his daughter

My partner and I have been together almost 6 years. His daughter is almost 9 and has lived with us since she was 3 1/2 coz her mum is a crazy druggy. She is a really bad kid and has been violent with my 6 yr old son and our daughter who Is 3 yr old. She has tortured animals, breaks the other childrens toys, talks nasty, vandalised the school bathroom by slandering a girl she didn't even know. Most recently, she damaged my mothers expensive coffee table by biting all along the edge of it. I have taken her to a psychologist and a peadiatrician over the years,both telling me she's fine. All the while my partner was a heavy dope smoker up until a year ago and Is now a heavy drinker. I am so frustrated with her and he does NOTHING except defend her and make excuses for her. He will only discipline her if I ask him too yet has no problem yelling at the other 2 children and giving them time outs. Tonight after the coffee table incident I came home hoping he had given her a punishment instead he got her take away dinner and she was sitting up giggling to her favourite TV show. Is it normal for me to hope things will change? Is it normal I'm fearful of her and her influence on my 2 children? I am so stressed and overwhelmed