From: clicliclic@freenet.de   
      
   "clicliclic@freenet.de" schrieb:   
   >   
   > I have been playing around with some old algebraic integrands in the   
   > new version 1.3.10 of FriCAS on the web interface.   
   >   
   > [...]   
   >   
   > The following integrand by Legendre is still evaluated to six complex   
   > logarithms:   
   >   
   > integrate(x/((4 - x^3)*sqrt(1 - x^3)), x)   
   >   
   > (((-3)^(1/2)+(-1))*((-1)/432)^(1/6)*log((((3888*x^5*(-3)^(1/2)   
   3888*x^5)*(((-1)/432)^(1/6))^5+(360*x^6+1440*x^3+(-1152))*(((-1)   
   432)^(1/6))^3+(((-6)*x^7+(-96)*x^4+48*x)*(-3)^(1/2)+(6*x^7+96*x^   
   +(-48)*x))*((-1)/432)^(1/6))*((-1)*x^3+1)^(1/2)+((((-864)*x^   
   7+(-864)*x^4+1728*x)*(-3)^(1/2)+(864*x^7+864*x^4+(-1728)*x))*(((   
   1)/432)^(1/6))^4+(((-36)*x^8+(-252)*x^5+288*x^2)*(-3)^(1/2)+((-3   
   )*x^8+(-252)*x^5+288*x^2))*(((-1)/432)^(1/6))^2+((-1)*x^9+(-66)*   
   ^6+72*x^3+(-32))))/(x^9+(-12)*x^6+48*x^3+(-64)))+(((-3)^(1/   
   2)+1)*((-1)/432)^(1/6)*log((((3888*x^5*(-3)^(1/2)+(-3888)*x^5)*(   
   (-1)/432)^(1/6))^5+((-360)*x^6+(-1440)*x^3+1152)*(((-1)/432)^(1/   
   ))^3+(((-6)*x^7+(-96)*x^4+48*x)*(-3)^(1/2)+((-6)*x^7+(-96)*x^4+4   
   *x))*((-1)/432)^(1/6))*((-1)*x^3+1)^(1/2)+(((864*x^7+864*x^   
   4+(-1728)*x)*(-3)^(1/2)+(864*x^7+864*x^4+(-1728)*x))*(((-1)/432)   
   (1/6))^4+((36*x^8+252*x^5+(-288)*x^2)*(-3)^(1/2)+((-36)*x^8+(-25   
   )*x^5+288*x^2))*(((-1)/432)^(1/6))^2+((-1)*x^9+(-66)*x^6+72*x^3+   
   -32))))/(x^9+(-12)*x^6+48*x^3+(-64)))+((-2)*((-1)/432)^(1/6)   
   *log(((7776*x^5*(((-1)/432)^(1/6))^5+((-360)*x^6+(-1440)*x^3+115   
   )*(((-1)/432)^(1/6))^3+(12*x^7+192*x^4+(-96)*x)*((-1)/432)^(1/6)   
   *((-1)*x^3+1)^(1/2)+(((-1728)*x^7+(-1728)*x^4+3456*x)*(((-1)/432   
   ^(1/6))^4+(72*x^8+504*x^5+(-576)*x^2)*(((-1)/432)^(1/6))^2+(   
   (-1)*x^9+(-66)*x^6+72*x^3+(-32))))/(x^9+(-12)*x^6+48*x^3+(-64)))   
   (2*((-1)/432)^(1/6)*log((((-7776)*x^5*(((-1)/432)^(1/6))^5+(360*   
   ^6+1440*x^3+(-1152))*(((-1)/432)^(1/6))^3+((-12)*x^7+(-192)*x^4+   
   6*x)*((-1)/432)^(1/6))*((-1)*x^3+1)^(1/2)+(((-1728)*x^7+(-   
   1728)*x^4+3456*x)*(((-1)/432)^(1/6))^4+(72*x^8+504*x^5+(-576)*x^   
   )*(((-1)/432)^(1/6))^2+((-1)*x^9+(-66)*x^6+72*x^3+(-32))))/(x^9+   
   -12)*x^6+48*x^3+(-64)))+(((-1)*(-3)^(1/2)+(-1))*((-1)/432)^(1/6)   
   log(((((-3888)*x^5*(-3)^(1/2)+3888*x^5)*(((-1)/432)^(1/6))^   
   5+(360*x^6+1440*x^3+(-1152))*(((-1)/432)^(1/6))^3+((6*x^7+96*x^4   
   (-48)*x)*(-3)^(1/2)+(6*x^7+96*x^4+(-48)*x))*((-1)/432)^(1/6))*((   
   1)*x^3+1)^(1/2)+(((864*x^7+864*x^4+(-1728)*x)*(-3)^(1/2)+(864*x^   
   +864*x^4+(-1728)*x))*(((-1)/432)^(1/6))^4+((36*x^8+252*x^5+(   
   -288)*x^2)*(-3)^(1/2)+((-36)*x^8+(-252)*x^5+288*x^2))*(((-1)/432   
   ^(1/6))^2+((-1)*x^9+(-66)*x^6+72*x^3+(-32))))/(x^9+(-12)*x^6+48*   
   ^3+(-64)))+((-1)*(-3)^(1/2)+1)*((-1)/432)^(1/6)*log(((((-3888)*x   
   5*(-3)^(1/2)+(-3888)*x^5)*(((-1)/432)^(1/6))^5+((-360)*x^6+(   
   -1440)*x^3+1152)*(((-1)/432)^(1/6))^3+((6*x^7+96*x^4+(-48)*x)*(-   
   )^(1/2)+((-6)*x^7+(-96)*x^4+48*x))*((-1)/432)^(1/6))*((-1)*x^3+1   
   ^(1/2)+((((-864)*x^7+(-864)*x^4+1728*x)*(-3)^(1/2)+(864*x^7+864*   
   ^4+(-1728)*x))*(((-1)/432)^(1/6))^4+(((-36)*x^8+(-252)*x^5+   
   288*x^2)*(-3)^(1/2)+((-36)*x^8+(-252)*x^5+288*x^2))*(((-1)/432)^   
   1/6))^2+((-1)*x^9+(-66)*x^6+72*x^3+(-32))))/(x^9+(-12)*x^6+48*x^   
   +(-64))))))))/36   
   >   
   > ... even though a real expression for the antiderivative exists:   
   >   
   > INT(x/((4 - x^3)*SQRT(1 - x^3)), x) =   
   > 2^(1/3)/18*(ATANH(SQRT(1 - x^3))   
   >  - 3*ATANH((1 + 2^(1/3)*x)/SQRT(1 - x^3))   
   >  - SQRT(3)*ATAN((2^(1/3) - 2^(2/3)*x - x^2)   
   > /(SQRT(3)*2^(1/3)*SQRT(1 - x^3))))   
   >   
   > Can't these complex logarithms be broken down similar to those for   
   > integral 5.66 (#401) from the Timofeev suite?   
   >   
   > [...]   
   >   
      
   Sadly, this observation remains true for the current version 1.3.12 of   
   FriCAS on the web interface. For a similar pseudo-elliptic integrand   
   given (in essence) by Euler, FriCAS 1.3.12 returns:   
      
   integrate(x/((8 + x^3)*sqrt(1 - x^3)), x)   
      
   (2*log((((-6)*x+6)*((-1)*x^3+1)^(1/2)+((-1)*x^3+12*x^2+6*x+10))/   
   x^3+6*x^2+12*x+8))+((-1)*log((((-6)*x^4+60*x^3+(-144)*x^2+24*x+1   
   0)*((-1)*x^3+1)^(1/2)+(x^6+(-24)*x^5+114*x^4+(-128)*x^3+(-48)*x^   
   +(-24)*x+136))/(x^6+(-6)*x^5+24*x^4+(-56)*x^3+96*x^2+(-96)*   
   x+64))+(-2)*3^(1/2)*atan(((x^2+10*x+(-8))*3^(1/2)*((-1)*x^3+1)^(   
   /2))/(9*x^3+9*x^2+(-18)*x))))/108   
      
   So, FriCAS thus succeeds in obtaining a real antiderivative here; a   
   simpler evaluation in real terms is:   
      
   INT(x/((8 + x^3)*SQRT(1 - x^3)), x) =   
   1/18*ATANH((1 - x)^2/(3*SQRT(1 - x^3)))   
    - 1/18*ATANH(SQRT(1 - x^3)/3)   
    - 1/(6*SQRT(3))*ATAN(SQRT(3)*(1 - x)/SQRT(1 - x^3))   
      
   Could the six complex logarithms be due to FriCAS somehow failing to   
   factorize a sextic of the type z^6 - a^6 in evaluating a sum over its   
   roots?   
      
   Anyway, FriCAS 1.3.12 fails on this integrand as well:   
      
   integrate((2 + x^3)/((4 - x^3)*sqrt(1 - x^3)), x)   
      
   while it may succeed again on this one:   
      
   integrate((4 - x^3)/((8 + x^3)*sqrt(1 - x^3)), x)   
      
   ... I just haven't tried it yet.   
      
   Martin.   
      
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