From: nma@12000.org   
      
   On 11/6/2013 6:16 PM, Albert Rich wrote:   
   > I am currently transcribing the 109 integration examples in Chapter 8 of   
   Timofeev's book.   
   >However I am unable to figure out what Timofeev intended for examples #66 and   
   #69 on   
   >page 366.  The integrands for #66 and #69 appear to be   
   >   
   > (cosh(x)^2 - sinh(x)^3) / (cosh[x)^3 + sinh(x)^3)   
   >   
   > and   
   >   
   > (tanh(x)-cosh(2x))*cosh(x)/((sinh(2x)+sinh(x)^2)*sqrt(sinh(2*x)))   
   >   
   > respectively, but the antiderivatives shown for them are not valid.   
   >Can anybody out there help me out?   
   >   
   > Albert   
   >   
      
   Yes, there is an error/typo somewhere. difference between   
   the derivative of the anti derivative and the integrand is  not   
   zero and not even linear constant difference. This is from   
   the 1948 edition. May be there is a newer edition than this   
   to check?   
      
   #66   
   restart;   
   integrand:=(cosh(x)^2 - sinh(x)^3) / (cosh(x)^3 + sinh(x)^3):   
   anti:= 1/(3*(1+tanh(x)))+ 4/(3*sqrt(3))*arctan( (2*tanh(x)-1)/sqrt(3)):   
   check:=simplify(integrand-diff(anti,x));   
      
     (1/3)*(3*cosh(x)^3-6*cosh(x)^2*sinh(x)+3*cosh(x)^2-4*cosh(x)   
     +2*sinh(x)) /(cosh(x)^3+cosh(x)^2*sinh(x)-sinh(x))   
      
   plot(%,x=-Pi..Pi);   
      
   #69   
   restart;   
   integrand:=(tanh(x)-cosh(2*x))*cosh(x)/((sinh(2*x)+sinh(x)^2)*sqrt(sinh(2*x))):   
   anti:= 1/sqrt(2*tanh(x))+sqrt(2)/6*log( (1-sqrt(tanh(x)))/(1+sqrt(tanh(x))) )   
   +sqrt(2)*arctan(tanh(x))+1/6 * arctan(sqrt(tanh(x)/2)):   
   check:=simplify(integrand-diff(anti,x)); %too large to show   
   plot(%,x=-Pi..Pi);   
      
   --Nasser   
      
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