[continued from previous message]   
      
   remaining cube roots cancel and the integrand becomes a rational   
   function of u. Carrying out that algebra (routine but lengthy) yields   
   the elementary antiderivative (with constant C):   
      
   I(x) = (1/36) \C2B7\ ln \E28E9B\ x^2 \E28892\ x\C2B7\(1 + 8t^2)^{1/3}/(1   
   \E28892\ t^2)^{1/3} + (1 + 8t^2)^{2/3}/(1 \E28892\ t^2)^{2/3} \E28E9E\   
   \E28892\ (1/6\E2889A\3) \C2B7\ arctan \E28E9B\ (2x\C2B7\(1 +   
   8t^2)^{1/3}/(1 \E28892\ t^2)^{1/3} \E28892\ (1 + 8t^2)^{2/3}/(1 \E28892\   
   t^2)^{2/3}) /(\E2889A\3 \C2B7\ (1 + 8t^2)^{1/3}/(1 \E28892\ t^2)^{1/3})   
   \E28E9E\   
      
   C,   
      
   where t = \E2889A\((x^3\E28892\1)/(x^3+8)) and the cube-root expressions   
   are to be interpreted as the real principal cube roots (branch chosen   
   consistently on the domain of interest).   
      
   This combination of logarithm and arctangent (with algebraic arguments   
   built from the elementary substitutions above) is the explicit   
   elementary primitive Goursat obtained for the integrand. The form can be   
   simplified or rewritten in many equivalent algebraic ways by clearing   
   the cube-root factors using (2); the displayed expression is one   
   convenient explicit antiderivative. If you want a fully expanded form   
   with no nested radicals (only rational functions, logs and arctan of   
   rational expressions), I can expand and simplify symbolically for the   
   real branch you care about (give the x\E28091\interval).   
      
   = = =   
      
   I should have asked for "a short summary of [Goursat's] reduction" too.   
      
   Martin :).   
      
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