On 6/8/2013 10:17 AM, email@example.com wrote: > I haven't checked the evaluations in the file systematically, but a new > look has revealed further possibilities for improvement. > > The present solutions of problems #21 and #22 from Charlwood's appendix, > INT(x^3*ASIN(x)/SQRT(1-x^4), x) and INT(x^3*ASEC(x)/SQRT(x^4-1), x), can > be written as: > > 1/4*(x*SQRT(1-x^4)/SQRT(1-x^2) > + LN(1-x^2) - LN(-x + x^3 + SQRT(1-x^2)*SQRT(1-x^4))) > - 1/2*SQRT(1-x^4)*ASIN(x)
How about ((asinh(x))/4) - ((sqrt(1 - x^4) * asin(x))/2) + ((x * sqrt(x^2 + 1))/4)
from Macsyma. Which has the advantage of using no logs, just asin() and asinh(). Since the integrand has asin, I think it is nice.
I have tried only a few others in the file, and Macsyma stumbled on one or two of them. I am not inclined generally to spend my time to see if the Macsyma solutions are smaller, neater, continuous etc, and maybe it is of no interest since Macsyma is not easily available. But then neither is Derive.. I suppose I can run the file through Macsyma if someone else is willing to look at the results :)