```Date: Jun 17, 2013 1:49 PM
Author: Nasser Abbasi
Subject: Re: The Charlwood Fifty

On 6/17/2013 11:11 AM, clicliclic@freenet.de wrote:>> 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 noticed that last night when I made a plot of them to compare. Here isthe plothttp://12000.org/tmp/061713/no_5.pngI might add a link then next to each given optimal entry inthe table showing a plot of the antiderivate, will be easy to add.> I usually accept logarithmic evaluations like INT(1/x, x) = LN(x), which> can be complex where the integrand is real (here for x < 0). I think> that users (e.g. calculus students) who need this integral from x = -2> to x = -1, say, should be able to accept that constants involving some> formal quantity #i appear which drop out of the final result.>> Martin.>Thanks for the information, this helped.On a related point, would you please help me understand howfree version of reduce transformed    arcsin(x)*log(x)to   arcsin( sin(g0) ) *  cos(g0)  * log( sin(g0) )by replacing x with sin(g0).i.e Where does cos(g0) term come from in the above transformation?Here is a link to the reduce trace for integral #1, which itcould not do btw. And the above was the first step in the process.http://12000.org/my_notes/ten_hard_integrals/reduce_logs/1/HTML/trace_1.htmlthanks,--Nasser
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