The Hermite reduction's system is solved in one row order whatever the spelling

BREAKING-CHANGES.md

@@ -326,6 +326,21 @@ improper fraction is declined before the first division rather than after the la
 | `"(1 - b*x^2)^3/(c*(1 - b*x^2) + a*d*x^2)^3".ToEntity().Integrate("x")` | no answer within a minute | the antiderivative |
 | `"(a + b*x)^(5/2)/(c + d*x)^4".ToEntity().Integrate("x")` | `integral(...)` | the antiderivative |
 
+### The Hermite reduction's system is solved in one order whatever the spelling
+
+**Faster, and not a different value.** The reduction of a rational integrand with a repeated factor
+below the bar solves one linear system for the rational part, and took its rows in the order the
+powers of `x` were met, which follows how the factors are written. With symbols in the coefficients
+the elimination's pivots followed that order, and in one order the solution came out as quotients of
+polynomials of the thirty-sixth degree in the symbols that nothing reduced:
+`x/((1 + x^2)^3 (2 a x + b (x^2 + 1)))` took 42 s, and with `(x^2 + 1)^3` below the bar 1 s. The rows
+are ordered by their power of `x` now. Where the order mattered, the antiderivative's coefficients can
+come out reduced where they were not.
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x/((1 + x^2)^3*(2*a*x + b*(x^2 + 1)))".ToEntity().Integrate("x")` | `integral(...)` | the antiderivative, in about half a second, as with `(x^2 + 1)^3` |
+
 ### A symbolic parameter no longer stops a rational integrand being integrated
 
 `1/(8 + x^3)` and `1/(16 - x^4)` are answered at once. `1/(a^3 + x^3)` and `1/(a^4 - x^4)` were not,

Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs

@@ -12766,7 +12766,12 @@ private static Entity SquarefreePartAsWritten(Entity denominator, Entity.Variabl
                 ? PolynomialProduct(PolynomialProduct(abovePoly, dSquared), qSquared)
                 : PolynomialProduct(PolynomialProduct(abovePoly, dSquared), squarefreePoly);
 
-            var powers = columns.SelectMany(c => c.Keys).Concat(targetRead.Keys).Distinct().ToList();
+            // The rows by their power of x, not in the order the dictionaries met them: that order
+            // is the spelling's, and the elimination's pivots follow it. `(1 + x^2)^3` below the bar
+            // where `(x^2 + 1)^3` is written left the solution in coefficients of the thirty-sixth
+            // degree in the symbols that nothing cancelled, and the logarithmic part they made was
+            // forty seconds of declining; in the one order both are under a second.
+            var powers = columns.SelectMany(c => c.Keys).Concat(targetRead.Keys).Distinct().OrderBy(power => power).ToList();
             var width = columns.Count;
             var matrix = new Entity[powers.Count][];
             var rhs = new Entity[powers.Count];

Sources/Tests/UnitTests/Calculus/RationalFactorsBesideASymbolicQuadraticTest.cs

@@ -124,5 +124,18 @@ private static void AnswersAndDifferentiatesBack(string integrand)
         [InlineData("tan(x)^4/(a+b*csc(x))")]
         [InlineData("sin(x)^2/(a+b*cos(x))")]
         public void TheNeighboursStayAnswered(string integrand) => AnswersAndDifferentiatesBack(integrand);
+
+        /// <summary>
+        /// The same integrand with `1 + x^2` written either way round. The Hermite reduction's
+        /// system took its rows in the order its dictionaries met the powers, which is the
+        /// spelling's, and with `(1 + x^2)^3` below the bar the solution came out in coefficients
+        /// of the thirty-sixth degree in the symbols that nothing cancelled: forty seconds of
+        /// declining their logarithmic part before the answer, where `(x^2 + 1)^3` took one.
+        /// </summary>
+        [Theory]
+        [InlineData("x/((1+x^2)^3*(2*a*x+b*(x^2+1)))")]
+        [InlineData("x/((x^2+1)^3*(2*a*x+b*(x^2+1)))")]
+        [InlineData("4*x*(1-x^2)^2/((1+x^2)^3*(2*a*x+b*(1+x^2)))")]
+        public void EitherSpellingOfOnePlusTheSquare(string integrand) => AnswersAndDifferentiatesBack(integrand);
     }
 }