On 6/17/2013 11:11 AM, email@example.com 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 is the plot
I might add a link then next to each given optimal entry in the 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 how free version of reduce transformed
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 it could not do btw. And the above was the first step in the process.