From: samuel.thomas.blake@gmail.com   
      
   On Saturday, February 10, 2024 at 4:23:26 AM UTC+11, nob...@nowhere.invalid   
   wrote:   
   > I have been playing around with some old algebraic integrands in the    
   > new version 1.3.10 of FriCAS on the web interface.    
   >    
   > Sam Blake's pseudo-elliptic of April 2020 still gives:    
   >    
   > integrate((x^4 - 1)*(x^4 + x^2 + 1)*sqrt(-x^4 + x^2 - 1)    
   > /(x^4 + 1)^3, x)    
   >    
   > >> Error detected within library code:    
   > catdef: division by zero    
   >    
   > perhaps because the radicand is negative everywhere on the real axis.    
   >    
   > And an older and presumably truly elliptic case still fails:    
   >    
   > integrate((5*x - 9*sqrt(6) + 26)    
   > /((x^2 - 4*x - 50)*sqrt(x^3 - 30*x - 56)), x)    
   >    
   > >> Error detected within library code:    
   > catdef: division by zero    
   >    
   > in the same manner, although the radicand is cubic here.    
   >    
   > 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?    
   >    
   > And for the next integrand, FriCAS still produces unreasonable integers    
   > in an arc tangent's argument:    
   >    
   > integrate(1/((x + 1)*(x^3 + 2)^(1/3)), x)    
   >    
   > (log(((21*x^4+(-6)*x^3+(-96)*x^2+(-60)*x+12)*((x^3+2)^(1/3))^2   
   (21*x^5+(-48)*x^3+102*x^2+228*x+96)*(x^3+2)^(1/3)+(22*x^6+6*x^5+   
   -48)*x^4+44*x^3+24*x^2+(-192)*x+(-140)))/(x^6+6*x^5+15*x^4+20*x^   
   +15*x^2+6*x+1))+2*3^(1/2)*atan(((   
   98966744593197647869364591874*x^4+190053406517364372745124029472   
   x^3+(-642339750020464731448133545632)*x^2+(-17643824508924025093   
   1037276448)*x+(-1072244631963565627440642667696))*3^(1/2)*((x^3+   
   )^(1/3))^2+((-45228634350310035870300951616)*x^5+(-   
   453545129950193664973324584892)*x^4+(-72617572249914718646544536   
   320)*x^3+735314591615271415729365586328*x^2+22308428093000003224   
   9227290544*x+1190118508012558386973005239952)*3^(1/2)*(x^3+2)^(1   
   3)+(93292570833559435663132301885*x^6+   
   382151535711085278859235047618*x^5+67392407422440877295962538479   
   *x^4+889426563183087468015580290048*x^3+888876515195959220955879   
   45824*x^2+351260598258508240019971964880*x+(-4767400099559721105   
   816884304))*3^(1/2))/(236716304443694165237125394649*x^6+   
   1013240117509374668590043803350*x^5+4679685832817576368300821292   
   *x^4+(-2686291575945300326054363894472)*x^3+10850035867214310866   
   8600126056*x^2+7625406903034897531937916271008*x+466444586047000   
   276943457906640)))/12    
   >    
   > ... while the antiderivative can in fact be compactly stated as:    
   >    
   > INT(1/((x + 1)*(x^3 + 2)^(1/3)), x) =    
   > 1/12*(- 3*LN((x^3 + 2)^(1/3) - x)    
   > + 2*SQRT(3)*ATAN(1/SQRT(3)*(1 + 2*x/(x^3 + 2)^(1/3))))    
   > - 1/4*(LN((x + 2)^3 - (x^3 + 2))    
   > - 3*LN((x + 2) - (x^3 + 2)^(1/3))    
   > + 2*SQRT(3)*ATAN(1/SQRT(3)*(1 + 2*((x + 2)/(x^3 + 2)^(1/3)))))    
   >    
   > If the unreasonable numbers cannot be avoided earlier, they could at    
   > least be removed by subtracting an arc tangent for a suitably chosen    
   > value of x; both x = infinity and x = -2^(1/3) turn out to work well.    
   >    
   > Finally I find that FriCAS version 1.3.10 still cannot solve:    
   >    
   > integrate((3*x + 2)/((x + 6)*(9*x - 2)*(3*x^2 + 4)^(1/3)), x)    
   >    
   > >> Error detected within library code:    
   > integrate: implementation incomplete (residue poly has multiple    
   > non-linear factors)    
   >    
   > as first presented in the thread "Risch integrator troubles" of    
   > Autumn/Winter 2019/2020. Why does this one remain too hard for an    
      
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