When evaluating the Bode integral, I am not getting even close to the ideal result (something negative)?

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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!