Albert Rich schrieb:   
   >   
   > On Tuesday, March 25, 2014 7:03:13 AM UTC-10, clicl...@freenet.de wrote:   
   > >   
   > > This must be because you are keeping everybody busy struggling along   
   > > behind Rubi, trying to catch up ...   
   >   
   > Thanks for stroking my ego, but I can't believe the legions of PhDs   
   > and grad students behind what are advertised as the world's most   
   > sophisticated computer algebra systems can't keep up with a beekeeper   
   > in Germany and a beach bum in Hawaii...   
   >   
      
   Now you are hiding your light under a bushel. To cheer you up I took   
   another look at the hypergeometric evaluations for integrals from   
   Chapter 8 of Timofeev's book.   
      
   - Example 3.n (p. 346) avoiding the 2F1 branch cut:   
      
   INT((#e^x-#e^(-x))^n,x)=1/(2*(n+1))*#e^x*(#e^x-#e^(-x))^(n+1)*F2~   
   1(1,1+n/2,2+n,1-#e^(2*x))   
      
   - Example 5b.n (p. 346) avoiding the 2F1 branch cut:   
      
   INT((a^(k*x)-a^(l*x))^n,x)=1/((k-l)*(n+1)*LN(a))*a^(-l*x)*(a^(k*~   
   x)-a^(l*x))^(n+1)*F21(1,1+k*n/(k-l),2+n,1-a^((k-l)*x))   
      
   - Example 6a.n (p. 346) avoiding the 2F1 branch cut (F21(-a^(m*x))   
   cannot be used here, compare example 5a.n for l=0):   
      
   INT((1+a^(m*x))^n,x)=1/(m*n*LN(a))*a^(-m*x)*(1+a^(m*x))^(n+1)*F2~   
   1(1,1,1-n,-a^(-m*x))   
      
   - Example 14 (p. 347) alternatives avoiding the 2F1 branch cut in   
   dependence of SIGN(a/b):   
      
   INT((a+b*#e^(n*x))^(r/s),x)=s/(n*r*b)*#e^(-n*x)*(a+b*#e^(n*x))^(~   
   r/s+1)*F21(1,1,1-r/s,-a/b*#e^(-n*x))=-s/(n*(r+s)*a)*(a+b*#e^(n*x~   
   ))^(r/s+1)*F21(1,1+r/s,2+r/s,1+b/a*#e^(n*x))   
      
   - Example 17 (p. 348) avoiding the 2F1 branch cut:   
      
   INT((#e^(7*x)-3)^(2/3)/#e^(2*x),x)=1/(35*3^(2/7))*(#e^(7*x)-3)^(~   
   5/3)*F21(9/7,5/3,8/3,1-1/3*#e^(7*x))   
      
   - Example 35 (p. 355) made more compact:   
      
   INT(#e^(m*x)/COS(x)^3,x)=8*(#e^((m+3*#i)*x)/(m+3*#i))*F21(3,(3-#~   
   i*m)/2,(5-#i*m)/2,-#e^(2*#i*x))   
      
   - Example 38 (p. 356) made more compact (compare example 36b):   
      
   INT(#e^x*(1+SIN(x))/(1-COS(x)),x)=#e^x*SIN(x)/(1-COS(x))+2*INT(#~   
   e^x/(1-COS(x)),x)=#e^x*SIN(x)/(1-COS(x))-2*(1-#i)*#e^((1+#i)*x)*~   
   F21(2,1-#i,2-#i,#e^(#i*x))   
      
   - Example 40 (p. 356) made more compact (compare example 36a):   
      
   INT(#e^x*(1-SIN(x))/(1+COS(x)),x)=-#e^x*SIN(x)/(1+COS(x))+2*INT(~   
   #e^x/(1+COS(x)),x)=-#e^x*SIN(x)/(1+COS(x))+2*(1-#i)*#e^((1+#i)*x~   
   )*F21(2,1-#i,2-#i,-#e^(#i*x))   
      
   - Example 41 (p. 356) made more compact (compare example 36d):   
      
   INT(#e^x*(1-COS(x))/(1-SIN(x)),x)=-#e^x*COS(x)/(1-SIN(x))+2*INT(~   
   #e^x/(1-SIN(x)),x)=-#e^x*COS(x)/(1-SIN(x))+2*(1+#i)*#e^((1+#i)*x~   
   )*F21(2,1-#i,2-#i,-#i*#e^(#i*x))   
      
   - Example 43 (p. 356) made more compact (compare example 36c):   
      
   INT(#e^x*(1+COS(x))/(1+SIN(x)),x)=#e^x*COS(x)/(1+SIN(x))+2*INT(#~   
   e^x/(1+SIN(x)),x)=#e^x*COS(x)/(1+SIN(x))-2*(1+#i)*#e^((1+#i)*x)*~   
   F21(2,1-#i,2-#i,#i*#e^(#i*x))   
      
   I didn't look at the elementary evaluations in Chapter 8.   
      
   Martin.   
      
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