MaplePrimes - answers and comments on Question, Integrating cos(sin(theta))

non-elementary

Robert Israel — Tue, 23 Sep 2008 22:00:36 Z

Maple returns int(cos(sin(theta)),theta) unevaluated. Since this is the "transcendental" case, for which I believe Maple's implementation of the Risch algorithm is pretty much complete, I'm pretty confident that Maple is correct in that this function has no elementary antiderivative.

definite or indefinite?

edgar — Fri, 26 Sep 2008 17:31:56 Z

Certain definite integrals of this kind are known... int(cos(a*sin(t)),t=0..Pi/2); (1/2)*Pi*BesselJ(0, a) --- G A Edgar

However

jakubi — Fri, 26 Sep 2008 19:20:10 Z

this knowledge is somewhat limited:

int(cos(a*sin(t)),t=0..Pi/2);
                         1/2 Pi BesselJ(0, a)

int(cos(a*sin(t)),t=0..Pi);

                           Pi BesselJ(0, a)

int(cos(a*sin(t)),t=0..2*Pi);

                          2 Pi
                         /
                        |
                        |      cos(a sin(t)) dt
                        |
                       /
                         0

Apparently, the periodicity and symmetries of the integrand are not used in the latter case.

periodicity and symmetries

John May — Fri, 26 Sep 2008 22:54:06 Z

Many of the internal integrators do check for periodicity and symmetries, but apparently they are not catching compositions of trig functions. I will look into beefing this up a bit.

John