With the current CVS version and in a fresh Maxima this problem is not present:

Maxima version: 5.18post

Maxima build date: 0:14 7/27/2009

host type: i686-pc-linux-gnu

lisp-implementation-type: CLISP

lisp-implementation-version: 2.44.1 (2008-02-23) (built 3436700604) (memory 3457635278)

(%i4) integrate(log(sec(x)),x);

(%o4) (x*log(sin(2*x)^2+cos(2*x)^2+2*cos(2*x)+1)

+2*%i*x*atan2(sin(2*x),cos(2*x)+1)-%i*li[2](-%e^(2*%i*x))-%i*x^2)

/2

+x*log(sec(x))

I have not checked the result.

Dieter Kaiser