From: maple@rogers.com   
      
   On Saturday, March 3, 2018 at 12:28:41 AM UTC-5, Nasser M. Abbasi wrote:   
   > so far, Maple 2017.3 and Mathematica 11.2 give wrong results   
   > on this:   
   >   
   > integrand := cos(x)* sin(x)^2*cos(n*x);   
   >   
   > Now I wanted to integrate the above from 0..2PI   
   > with the assumption that n above is integer and   
   > positive.   
   >   
   > What does your CAS give for the integral?  hint:   
   > it should not be zero.  But....   
   >   
   > Maple:   
   > integrand := cos(x)* sin(x)^2*cos(n*x);   
   > assume(n,integer,n>0);   
   > int(integrand,x=0..2*Pi);   
   >   
   >             0   
   >   
   > Mathematica:   
   > Assuming[Element[n, Integers] && n > 0,   
   >              Integrate[integrand, {x, 0, 2 Pi}]]   
   >      0   
   >   
   > But this integral is not zero for all integer n.   
   >   
   > Table[Integrate[integrand, {x, 0, 2 Pi}], {n, 1, 5}]   
   >          {Pi/4,0,-Pi/4,0,0}   
   >   
   > I do not know how to use assumptions yet in Fricas   
   > to test this. Will try to learn how to.   
   >   
   > I really do not understand how advanced and mature CAS   
   > systems like Maple and Mathematica can get this wrong.   
   >   
   > Or may be it is by design. I do not know. Someone told   
   > me it is hard to program these exception cases into CAS   
   > and have it check for each special case.  Any thoughts?   
   >   
   > --Nasser   
      
      
   kernelopts(version);   
      Maple 2017.2, X86 64 LINUX, Jul 19 2017, Build ID 1247392   
      
   restart;   
   integrand := cos(x)* sin(x)^2*cos(n*x):   
   assume(n,integer,n>0);   
   int(integrand,x=0..2*Pi,allsolutions);   
      
                        /  1   
                        |  - Pi         n = 1   
                        |  4   
                        |   
                       <    1   
                        | - - Pi        n = 3   
                        |   4   
                        |   
                        \   0         otherwise   
      
   restart;   
   integrand := cos(x)* sin(x)^2*cos(n*x):   
   int(integrand,x=0..2*Pi,allsolutions) assuming n::integer, n>0;   
      
                        /  1   
                        |  - Pi         n = 1   
                        |  4   
                        |   
                       <    1   
                        | - - Pi        n = 3   
                        |   4   
                        |   
                        \   0         otherwise   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)