On 6/9/2013 7:40 PM, Albert Rich wrote: > On Sunday, June 9, 2013 3:37:09 PM UTC-10, Richard Fateman wrote: > >> INT(x^3*ASEC(x)/SQRT(x^4-1), x) >> >> is >> >> ((a * (sqrt(x^2 - 1) * asec(x) - 1))/2) + ((log(((a + 1)/(a - >> 1))))/4) >> >> with a=sqrt(x^2+1). >> >> I have not investigated how this fairs with respect to maximal >> continuity. > > Unfortunately, that is not a valid antiderivative since subtracting > its derivative from the original integrand and substituting -2 for x > does not equal zero. > > Albert >
It checks out in Macsyma. In separate correspondence with Albert Rich, he suggested looking at how Macsyma defines the derivative of arcsecant. Indeed, Macsyma's definition differs by a sign for negative argument.
For Macsyma's definition, the integral is correct. For Maple or Mathematica or Maxima, the integral needs an extra abs().
I suppose the dlmf would provide the definitive form, but it seems to be offline at the moment. I'm guessing it agrees with Maxima, not Macsyma.