From: Albert_Rich@msn.com   
      
   On Wednesday, December 14, 2016 at 7:05:47 AM UTC-10, clicl...@freenet.de   
   wrote:   
   > Albert Rich schrieb:   
   > >   
   > > What would you recommend changing it to?   
   > >   
   >   
   > I don't necessarily recommend to change it: What is the most appropriate   
   > antiderivative should follow from your choice of design principles.   
   >   
   > In fact, I myself seem to have introduced antiderivatives of the   
   > ATANH(SIN(t)) type in my sci.math.symbolic post of Sun, 21 Apr 2013   
   > 18:20:53 +0200 to the thread "An independent integration test suite".   
   >   
   > This form of antiderivative is real on the entire real line just as the   
   > integrand is. The alternative LN(TAN(pi/4 + t/2)) is almost as simple,   
   > but its branch cuts in the complex plane are less disruptive while its   
   > imaginary part along the real line jumps between zero and pi.   
   >   
   > Many of the antiderivatives that exhibit logarithmic poles should admit   
   > two or more alternative forms like this.   
   >   
   > Martin.   
      
   A design principle I value highly is the symmetry between the trig and   
   hyperbolic functions.  For the antiderivatives of sec(x) and csc(x), Rubi   
   currently returns   
      
       arctanh(sin(x))  and  -arctanh(cos(x))   
      
   respectively.  For the antiderivatives of sech(x) and csch(x), it returns   
      
       arctan(sinh(x))  and  -arctanh(cosh(x))   
      
   respectively.  All nicely symmetric, except for there being just one arctan   
   and but three arctanhs...   
      
   I guess if a Rubi user changes the antiderivative of sec(x) to   
      
       log(tan(x/2+pi/4)),   
      
   he or she should also change the antiderivative of csc(x) to   
      
       log(tan(x/2)).   
      
   And for symmetry, should he or she also change the antiderivatives of sech(x)   
   and csch(x) to   
      
       i*log(tanh(x/2+pi*i/4))  and  log(tanh(x/2))   
      
   respectively, where i is the imaginary unit?   
      
   Albert   
      
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