"Nasser M. Abbasi" schrieb:   
   >   
   > 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?   
   >   
      
   Letting Derive 6.10 loose on your definite integral for symbolic   
   positive integer n:   
      
   n :epsilon Real(0, inf)   
      
   INT(COS(x)*SIN(x)^2*COS(n*x), x, 0, 2*pi)   
      
   2*n*SIN(2*pi*n)/((n + 1)*(1 - n)*(n + 3)*(n - 3))   
      
   For n from 1 to 8, this result for symbolic n evaluates to:   
      
   VECTOR(2*n*SIN(2*pi*n)/((n + 1)*(1 - n)*(n + 3)*(n - 3)), n, 1, 8)   
      
   [?, 0, ?, 0, 0, 0, 0, 0]   
      
   And now the integral itself for explicit integer n running from 1 to 8:   
      
   VECTOR(INT(COS(x)*SIN(x)^2*COS(n*x), x, 0, 2*pi), n, 1, 8)   
      
   [pi/4, 0, - pi/4, 0, 0, 0, 0, 0]   
      
   Perfect all around, I would say :).   
      
   Martin.   
      
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