Having trouble with this one.
Make the substitution .
You might want to check the details (as I rushed this) - I get something like .
Expand, integrate term-by-term and finally substitute back that (tedious, I know, but not too hard).
Ask for more help if you get stuck at any point.
Edit: There is a small mistake - you should spot it easily. A small detail that doesn't change the general game plan .....
I don't really understand either of your methods (you combined two steps?), but I used u-substitution then trig-substitution to get .
It appears that polymerase used the double-angle formula to get instead of ? I just wasn't sure what to do with that last term. I also made a small mistake with the coefficient of the first term, but I caught that the second time around.