I tried some Cosine Transforms found in the book 'Handbook of Integral
Equations' by  Andrei D. Polyanin and
Alexander V. Manzhirov into sage and I found that sage was not able to
perform these integrals.

I guess my questions are:
How powerful is the integrate() function or how well versed is the
integrate() in sage?
What is being done to improve the integrate() ?
Is there something that I could do to help this coming summer to
improve it?

I'm a junior math major at CSUN.  It seems that my summer is going to
be empty and I'd like to challenge myself in something.  It looks like
the integrate() could be a challenge.

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