In Response To 'Re: Bug in Integration'
Thank you very much for the nice solution of this problem. Actually, I was also thinking about the singularities at Pi/4's. Then I realized that this issue is addressed in Mathematica by using PrincipalValue-> True option.
I used this option and I got the same result as yours. However, it took 677 sec for mathematica to solve it.
Actually that integral was only the first part of the bigger expression. I was trying to separate the expressions to solve the integral since the second part varies slowly respect to this first part. However, the result was not satisfactory, so I ended up to use Taylor expansion on the whole expression and solve the problem.