When evaluating the Bode integral, I am not getting even close to the ideal result (something negative)?
This can be possible if you happen to have a poor design. The sensitivity function will not die down at very high frequencies. In this case, you can either try to extend the upper limit of your integral or use a different design with larger phase margin.
Also remember that the logarithm in the integrand is to the base \(e\,\), not 10!