can someone check this calculus problem for me?

This was on a test and this is one of the problems i got "wrong". I think that my calculation is right, because of the formula she gave us to do this one.
the formula is integral(sec^n(x)dx )= 1/(n-1)sec^(n-1)xtanx + (n-2)/(n-1)integral(sec^(n-2)xdx)
integrate sec^5(x)
I did this problem the correct way:First i did this formula twice since It gave me the integral sec^3 x. then i integrated sec(x) which is given at the end of the second cycle and it gave me the answer. I've tried this problem many times.
Here is my answer and it makes no sense to me why it is wrong. a 1/2 coefficient was counted wrong but i dont get why.
1/4sec^4(x)tan(x)+3/8sec^2(x)tan(x)+1/2ln|sec(x)+tan(x)| +c.
the last coefficent(1/2ln|sec(x) + tan(x) | was counted wrong andit makes no sense why because the formula ends up putting n-1/n-2 before the last integral. and n is equal to three in that part. can someone tell me if i'm right or the professor is right? thanks.

 

thanks