From: fateman@cs.berkeley.edu   
      
   On 11/21/2016 8:35 PM, Nasser M. Abbasi wrote:   
   >   
   > It is interesting how different CAS outputs differ sometimes.   
   > I was trying to verify book result for int(sec(t),t) and this   
   > is what some CAS systems gave   
   >   
   > Mathematica:     ln(cos(t/2)+sin(t/2))-ln(cos(t/2)-sin(t/2))   
   > Rubi:            arctanh(sin(t))   
   > Fricas:          1/2 ( ln(1+sin(t)) - ln(1-sin(t)) )   
   > Maple & Maxima:  ln(sec(t)+tan(t))   
   >   
   > Text Book happens to give result shown by Maple&Maxima.   
   >   
   > Ofcourse all other results are also correct, I am sure,   
   > and these can all be converted to each others.   
   >   
   > Was Wondering if there is any preference, math-wise, to any one   
   > of the above results. From the Mathematica result, it seems   
   > to hint that it used the Tangent half-angle substitution,   
   > also called Weierstrass substitution.   
   >   
   > What does you CAS give to int(sec(t),t) if different from   
   > the above?   
   >   
   > --Nasser   
      
   after trigsimp, Maxima gives log((1+sin(x))/cos(x))   
   which, if numerically evaluated,   
     uses 3 function evaluations of popular functions.   
   same count as Fricas. Rubi's answer is most compact, but   
   arctanh is not such a popular function.   
   Mathematica, even after "Fullsimplify"   
   has more function evaluations.   
      
   Maxima gives for risch(sec(x),x)   
   (log(sin(x)^2+2*sin(x)+cos(x)^2+1)-log(sin(x)^2-2*sin(x)+cos(x)^2+1))/2   
   which can be simplified to one of the above by trigsimp and   
   logcontract to   
      
   -log(-(sin(x)-1)/(sin(x)+1))/2   
      
   which has only 2 distinct function evals.  Slightly better than Fricas'   
   form.   
      
   RJF   
      
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