On Friday, April 10, 2020 at 9:07:34 AM UTC+10, anti...@math.uni.wroc.pl wrote:   
   > samuel.thomas.blake@gmail.com wrote:   
   > >   
   > > Here's a couple of other error messages I found in 1.2.6 (on OSX).   
   > >   
   > > (428) -> integrate(((-1+2*x^4)*(32+7*x^2+64*x^4)*(8+35*x^4+3   
   *x^8)^(1/2))/(x^3*(20-x^2+79*x^4-2*x^6+80*x^8)),x)   
   > >   
   > > ? ?>> Error detected within library code:   
   > > ? ?"failed" of mode Union(Integer,"failed") cannot be coerced to mode   
   Integer   
   >   
   > That was a silly typo.  Thanks for report, I have now commit a fix.   
   >   
   > >   
   > > integrate(((-1+x^4)*(1+x^2+x^4)*(-1+x^2-x^4)^(1/2))/(1+x^4)^3,x);   
   > >   
   > > ?? >> Error detected within library code:   
   > > ?? catdef: division by zero   
   > >   
   > >   
   > > integrate(((4+x^6)*(-4+x^4+2*x^6)*(32-14*x^4-32*x^6-4*x^8+7*   
   ^10+8*x^12)^(1/2))/(x^9*(-2+x^6)),x);   
   > > ?   
   > > ?? >> Error detected within library code:   
   > > ?? integrate: implementation incomplete (residue poly has multiple   
   non-linear factors)   
   >   
   > The two above are known problems, each will take more work to fix.   
   > Actually, the first one requires reorganization of integration   
   > routines.  The second one is king of open problem: known methods   
   > of solving may require prohibitivly long time and nobody knows   
   > of really efficient method.   
   >   
   > --   
   >                               Waldek Hebisch   
      
      
   In regards to the last integral which hits an "implementation incomplete"   
   message. Incase this is of any interest - Maple computes this integral using   
   their implementation of the Risch-Trager algorithm. A trace is given below.   
      
   $ maple   
       |\^/|     Maple 2018 (X86 64 LINUX)   
   ._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2018   
    \  MAPLE  /  All rights reserved. Maple is a trademark of   
    <____ ____>  Waterloo Maple Inc.   
         |       Type ? for help.   
   > infolevel[int]:=5:   
   > int(convert(((4+x^6)*(-4+x^4+2*x^6)*(32-14*x^4-32*x^6-4*x^8+7*   
   ^10+8*x^12)^(1/2))/(x^9*(-2+x^6)),RootOf),x);   
   int:   Beginning integration with _EnvContinuous=_EnvContinuous,   
   _EnvAllSolutions=_EnvAllSolutions, and _EnvCauchyPrincipalValue=_E\   
   nvCauchyPrincipalValue.   
   int/indef1:   first-stage indefinite integration   
   int/indef2:   second-stage indefinite integration   
   Norman:   enter Risch-Norman integrator   
   Norman:   exit Risch-Norman integrator   
   int/algrisch/int:   Risch/Trager's algorithm for algebraic function   
   int/algrisch/int:   entering integrator at time   .901   
   int/algrisch/int:   function field has degree   2   
   int/algrisch/int:   pol min is   _z^12*y^2+3*_z^12+114*_z^11-167   
   _z^10-1680*_z^9-4656*_z^8-7684*_z^7-8718*_z^6-7144*_z^5-4271*_z^4-\   
   1830*_z^3-535*_z^2-96*_z-8   integrand is   -(5*_z^12-74*_z^11-2   
   1*_z^10-656*_z^9-1264*_z^8-1860*_z^7-2062*_z^6-1704*_z^5-1035*_z^4   
   -450*_z^3-133*_z^2-24*_z-2)*_z/(_z+1)^9/(_z^6-6*_z^5-15*_z^4-20*   
   z^3-15*_z^2-6*_z-1)*y   
   int/algrisch/int:   computation of an integral basis: start time   .906   
   int/algrisch/int:   integral basis is    2, 2, {(1, 1) = 1, (2, 2) = _z^6},   
   datatype = anything, storage = rectangular, order =   
   Fortran_order, shape = []   
   int/algrisch/int:   computation of an integral basis: end time   .913   
   int/algrisch/int:   normalization at infinity: start time   .913   
   int/algrisch/int:   normal integral basis is   Matrix(2, 2, [[...],[...]],   
   datatype = anything)   
   int/algrisch/int:   discriminant of the basis is   3*_z^12+114*_   
   ^11-167*_z^10-1680*_z^9-4656*_z^8-7684*_z^7-8718*_z^6-7144*_z^5-\   
   4271*_z^4-1830*_z^3-535*_z^2-96*_z-8   
   int/algrisch/int:   normalization at infinity: end time   .925   
   int/algrisch/int:   genus of the function field   5   
   int/algrisch/int:   derivatives are   Matrix(2, 2, [[...],[...]], datatype =   
   anything)   
   int/algrisch/int:   denominator of the derivatives is   3*_z^12+   
   14*_z^11-167*_z^10-1680*_z^9-4656*_z^8-7684*_z^7-8718*_z^6-7144*_z   
   ^5-4271*_z^4-1830*_z^3-535*_z^2-96*_z-8   
   int/algrisch/int:   denominator of the integrand is   _z^20+3*_z   
   19-33*_z^18-287*_z^17-1113*_z^16-2751*_z^15-4837*_z^14-6363*_z^13-\   
   6417*_z^12-5003*_z^11-3003*_z^10-1365*_z^9-455*_z^8-105*_z^7-15*_z^6-_z^5   
   int/algrisch/int:   computation of the algebraic part: start time   .941   
   int/algrisch/algpart:   reducing factor of multplicity   9   
   int/algrisch/solvemodv1:   solving a linear system modulo   _z+1   
   int/algrisch/solvemodv1:   solutions are    [0, -1]   
   int/algrisch/algpart:   reducing factor of multplicity   8   
   int/algrisch/solvemodv1:   solving a linear system modulo   _z+1   
   int/algrisch/solvemodv1:   solutions are    [0, 2]   
   int/algrisch/algpart:   reducing factor of multplicity   7   
   int/algrisch/solvemodv1:   solving a linear system modulo   _z+1   
   int/algrisch/solvemodv1:   solutions are    [0, -1]   
   int/algrisch/algpart:   reducing factor of multplicity   6   
   int/algrisch/solvemodv1:   solving a linear system modulo   _z+1   
   int/algrisch/solvemodv1:   solutions are    [0, 0]   
   int/algrisch/algpart:   reducing factor of multplicity   5   
   int/algrisch/solvemodv1:   solving a linear system modulo   _z   
   int/algrisch/solvemodv1:   solutions are    [0, 1/2]   
   int/algrisch/solvemodv1:   solving a linear system modulo   _z+1   
   int/algrisch/solvemodv1:   solutions are    [0, 23/32]   
   int/algrisch/algpart:   reducing factor of multplicity   4   
   int/algrisch/solvemodv1:   solving a linear system modulo   _z   
   int/algrisch/solvemodv1:   solutions are    [0, 39/32]   
   int/algrisch/solvemodv1:   solving a linear system modulo   _z+1   
   int/algrisch/solvemodv1:   solutions are    [0, 39/32]   
   int/algrisch/algpart:   reducing factor of multplicity   3   
   int/algrisch/solvemodv1:   solving a linear system modulo   _z   
   int/algrisch/solvemodv1:   solutions are    [0, 0]   
   int/algrisch/solvemodv1:   solving a linear system modulo   _z+1   
   int/algrisch/solvemodv1:   solutions are    [0, 0]   
   int/algrisch/algpart:   reducing factor of multplicity   2   
   int/algrisch/solvemodv1:   solving a linear system modulo   _z   
   int/algrisch/solvemodv1:   solutions are    [0, 0]   
   int/algrisch/solvemodv1:   solving a linear system modulo   _z+1   
   int/algrisch/solvemodv1:   solutions are    [0, 0]   
   int/algrisch/int:   computation of the algebraic part: end time   1.001   
   int/algrisch/int:   coordinates of the algebraic part are   [0,    
   1/(_z+1)^8+2/(_z+1)^7-1/(_z+1)^6+(-7/32*_z+1/2)/(_z+1)^4/_z^4+39/   
   32/(_z+1)^3/_z^3]   
   int/algrisch/int:   coordinates of the transcendental part are   Vector(2,   
   [...], datatype = anything)   
   int/algrisch/int:   computation of the transcendental part: start time   1.002   
   int/algrisch/transcpar:   computing a basis for the residues at time   1.026   
   int/algrisch/residues:   computing a splitting field at time   1.027   
   int/algrisch/residues:   polynomial has degree   16   
      
   [continued in next message]   
      
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