Richard Fateman <firstname.lastname@example.org> wrote: > 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(). >
Numerical evaluation of this at negative numbers gives wrong value because standard branch choice for sqrt is wrong in this case: in the formula above taking positive branch of sqrt for x > 1 implies negative branch for x < -1. Using
1/(x^2*sqrt(1 - 1/x^2))
as formula for derivative avoids this problem. OTOH previous formula have its own advantages, so the choice is not so clear.
Concerning Macsyma result: for "branch correct" result Macsyma should use a = sqrt(x^4 - 1)/sqrt(x^2-1).