I notice that Timofeev (p. 422) gives an evaluation incompatible with the above: it doesn't differentiate back to something as simple as his integrand, so one may conclude he made the mistake rather than the typesetter.
As regards example #19 from Chapter 5, Timofeev gives a finite sum (p. 220) involving Sin[4*(m-p)*x] with p running from 0 to m-1 in place of your Hypergeometric2F1[1/2-m, 1/2+m, 3/2+m, Cos[x]^2] times elementary factors. Your hypergeometric series terminates when 2*m is an odd integer greater than -3; and restated as Hypergeometric2F1[1+2*m, 1, 3/2+m, Cos[x]^2] times different elementary factors, it terminates when m is a negative integer. Though not a polynomial, it is an elementary function for any non-negative integer m as well, for m=3 one has for instance
but at present I don't know how to express this in finite terms for arbitrary non-negative integer m. Unfortunately, non-negative integer m are what Timofeev had in mind here. On the other hand, I think the exponent m should not be restricted in the test suite.
As regards example #75 from Chapter 5, it seems that Timofeev meant to write (p. 273):
I notice that you write SIN(2*x)^(5/2) for his SQRT(SIN(2*x)^5); I concur that we should allow to modify integrands in this way if the piecewise constants become simpler in consequence. However, one shouldn't try to reinterpret radicals like SQRT(SIN(x)*COS(x)^2) in example #80, I think.
Can you point out where you couldn't map Timofeev's evaluations to your 'optimal' ones - modulo apparent misprints and any unavoidable piecewise constants? This would help focusing the attention of sci.math.symbolic readers; to compare all integrals and evaluations in your file against the book means an awful lot of work.