From: nma@12000.org   
      
   On 10/10/2013 1:21 AM, Albert Rich wrote:   
      
   >So I was delighted to discover the following identity that makes   
   >it easy to transform discontinuous antiderivatives into continuous ones:   
   >   
   > If  g(x) = arctan(a*tan f(x))  and   
   >h(x) = f(x) - arctan(cos f(x)*sin f(x) / (a/(1-a) + cos f(x)^2), <-- extra   
   ")" missing   
   >   
   >then g’(x) = h’(x).   
   >   
      
   Fyi, verified it:   
      
   Mathematica:   
      
   g[x_] := ArcTan[a*Tan [f[x]]]   
   h[x_] := f[x]-ArcTan[ Cos[f[x]]* Sin[f[x]] /(a/(1-a) + Cos[f[x]]^2)]   
   r1 = D[g[x], x];   
   r2 = D[h[x], x];   
   Simplify[r1 - r2]   
   (*  0 *)   
      
   Maple:   
      
   g(x) := arctan(a*tan(f(x)));   
   h(x) := f(x)-arctan(cos(f(x))*sin(f(x)) /(a/(1-a) + cos(f(x))^2));   
   simplify(  diff(g(x),x)-diff(h(x),x));   
        0   
      
   >   
   > The next release of Rubi will take advantage of these identities to produce   
   >continuous trig antiderivatives, thereby making the computation of definite   
   integrals easy and reliable.   
   >   
      
   Looking forward for Rubi 4.3 !   
      
   > Albert   
   >   
      
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