On Monday, June 17, 2013 6:11:36 AM UTC-10, clicl...@freenet.de wrote:
> > On 6/16/2013 11:30 AM, email@example.com wrote: > > > Oops, [problem #5] should have read: > > > > INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) = > > > = x/3 + 1/3*ATAN(SIN(x)*COS(x)*(COS(x)^2 + 1) > > > /(COS(x)^2*SQRT(COS(x)^4 + COS(x)^2 + 1) + 1)) > > Apart from the compactness of antiderivatives, as measured by leaf > counting, continuity on the real axis and absence of complex > intermediate results when evaluated on the real axis (which implies > absence of the imaginary unit) are important in my view, and usually > take precedence over compactness. > > Thus, my 45-leafed result is fully continuous along the real axis, > whereas the shorter ATAN alternative: > > INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) = > = - 1/3*ATAN(COT(x)*COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1)) > > as well as Albert's 37-leafed ASIN version: > > INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) = > = - ASIN(COS(x)^3)*SQRT(1 - COS(x)^6)*CSC(x) > /(3*SQRT(1 + COS(x)^2 + COS(x)^4)) > > jump at x = -pi, 0, pi, etc. This constitutes an unnecessary obstacle in > definite integration - imagine some quantity integrated along the path > of an orbiting spacecraft.
I agree continuity of antiderivatives trumps compactness. Your impressive, continuous result for problem #5 is in the revised Charlwood Fifty pdf file at