@@ -447,6 +447,18 @@ case is most of its length ([#718](https://github.com/asc-community/AngouriMath/
 | `"x^4/((a + b*x^2 + c*x^4)*sqrt(d + k*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 28,551 characters |
 | `"1/(x^2*(a + b*x^2 + c*x^4)*sqrt(d + k*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 10,194 characters |
 
+### A rational function past the twelfth degree is reduced by Hermite's method
+
+**Answers where there were none.** The integrator read a denominator of degree twelve at most, so a derivative over
+powers of `x^4 + x + 3` up to the fourth was declined, after seven to fifteen seconds. A rational function with numbers
+for coefficients over a denominator past the twelfth degree now takes Hermite's reduction, as long as what is left for the
+logarithms stays within twelve ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"3*(-47+228*x+120*x^2+19*x^3)/(3+x+x^4)^4+(42-320*x-75*x^2-8*x^3)/(3+x+x^4)^3+30*x/(3+x+x^4)^2".ToEntity().Integrate("x")` | `integral(...)` | 142 characters |
+| `"(-3+10*x+4*x^3-30*x^5)/(3+x+x^4)^3-3*(1+4*x^3)*(2-3*x+5*x^2+x^4-5*x^6)/(3+x+x^4)^4".ToEntity().Integrate("x")` | `integral(...)` | 142 characters |
+
 ### A power of a linear over a power of a quadratic is reduced a power at a time
 
 **Answers where there were none.** `sqrt(d + k x)/(a + b x + c x^2)^3` took thirty seconds for 26,437 characters on the
 
@@ -2621,6 +2621,50 @@ node is Powf(var radicand, Number.Rational r) && r is not Number.Integer && radi
         /// <see cref="RothsteinTrager"/>. After <see cref="SolveByPartialFractions"/>, so that
         /// everything that answers keeps the form it gives.
         /// </summary>
+        internal static Entity? SolveAHighDegreeRationalFunctionByHermite(Entity expr, Entity.Variable x)
+        {
+            // A sum of quotients over powers of one denominator, Rubi's spelling of a derivative,
+            // is one quotient for this. At the question asked only, which is where nothing else
+            // answered one.
+            if (!Integration.AnsweringTheQuestionAsked || expr.Vars.Any(symbol => symbol != x))
+                return null;
+            // Over the least common multiple of the terms' denominators, each base to its highest
+            // power, rather than their product: three terms over the fourth, third and second
+            // powers of one quartic are one quotient over its fourth power.
+            var terms = Sumf.LinearChildren(expr).Select(term => Functions.SingleQuotient.Of(term)).ToList();
+            var highest = new Dictionary<Entity, int>();
+            var powersOf = new List<Dictionary<Entity, int>>();
+            foreach (var (_, below) in terms)
+            {
+                var powers = new Dictionary<Entity, int>();
+                foreach (var factor in Mulf.LinearChildren(below))
+                {
+                    var (@base, whole) = factor is Powf(var b0, Number.Integer { EInteger: var w }) && w.Sign > 0 && w.CanFitInInt32()
+                        ? (b0, w.ToInt32Unchecked()) : (factor, 1);
+                    powers[@base] = powers.TryGetValue(@base, out var already) ? already + whole : whole;
+                }
+                foreach (var pair in powers)
+                    highest[pair.Key] = System.Math.Max(highest.TryGetValue(pair.Key, out var already) ? already : 0, pair.Value);
+                powersOf.Add(powers);
+            }
+            Entity denominator = Number.Integer.One;
+            foreach (var pair in highest)
+                denominator = denominator * MathS.Pow(pair.Key, Number.Integer.Create(pair.Value));
+            Entity numerator = Number.Integer.Zero;
+            for (var i = 0; i < terms.Count; i++)
+            {
+                Entity cofactor = Number.Integer.One;
+                foreach (var pair in highest)
+                {
+                    var missing = pair.Value - (powersOf[i].TryGetValue(pair.Key, out var have) ? have : 0);
+                    if (missing > 0)
+                        cofactor = cofactor * MathS.Pow(pair.Key, Number.Integer.Create(missing));
+                }
+                numerator = numerator + terms[i].Numerator * cofactor;
+            }
+            return RothsteinTrager.IntegrateAHighDegree(numerator, denominator, x);
+        }
+
         internal static Entity? SolveByRothsteinTrager(Entity expr, Entity.Variable x)
         {
             if (!TryReadAsQuotient(expr, out var numerator, out var denominator))
 
@@ -666,6 +666,9 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // the question asked only, and the polynomial term below writes `1/u` as `u^(-1)` and
             // asks that a level down: `1/(e^atanh(a x) (c - c/(a x))^(3/2))` reached it nowhere.
             if ((answer = IndefiniteIntegralSolver.SolveByWritingEachLinearOnce(expr, x, integrateByParts)) is { }) return answer;
+            // A rational function with numbers for coefficients over a denominator past the twelfth
+            // degree, by Hermite's reduction before anything searches: nothing below answered one.
+            if ((answer = IndefiniteIntegralSolver.SolveAHighDegreeRationalFunctionByHermite(expr, x)) is { }) return answer;
             // A power of a linear beside the derivative of a quadratic over a whole power of it, by
             // parts once, before anything takes the symbolic quadratic apart.
             if ((answer = IndefiniteIntegralSolver.SolveAPowerOfALinearBesideTheDerivativeOfAQuadraticOverItsPower(expr, x, integrateByParts)) is { }) return answer;
 
@@ -63,16 +63,44 @@ internal static class RothsteinTrager
         /// </summary>
         private const int MaxDegree = 12;
 
+        /// <summary>
+        /// The largest denominator degree read at all: Hermite's reduction is a few Diophantine
+        /// solves over the rationals, and what it leaves for the resultant is the squarefree part,
+        /// held to <see cref="MaxDegree"/>. Rubi's 1.3.1 <c>3 (19 x^3 + 120 x^2 + 228 x - 47)/(x^4 + x + 3)^4 + ...</c>
+        /// is a derivative over a sixteenth-degree denominator with nothing left for the logarithms,
+        /// and was declined after a minute and a half of searching elsewhere.
+        /// </summary>
+        private const int MaxReducedDegree = 24;
+
         /// <summary>
         /// The integral of <paramref name="numerator"/> over <paramref name="denominator"/>,
         /// both polynomials in <paramref name="x"/> with rational coefficients and the fraction
         /// proper; <see langword="null"/> where they are not, or where the answer does not
         /// differentiate back to the integrand at the sampled points.
         /// </summary>
+        /// <summary>
+        /// <see cref="Integrate"/>, for a denominator past <see cref="MaxDegree"/> only: what the
+        /// rest of the integrator never answered, asked before it searches.
+        /// </summary>
+        internal static Entity? IntegrateAHighDegree(Entity numerator, Entity denominator, Variable x)
+            => PolynomialFactoring.TryGetRationalCoefficients(denominator, x, 1, 1, MaxReducedDegree, out var belowCoefficients)
+               && RationalPolynomial.Create(belowCoefficients).Degree > MaxDegree
+                ? IntegrateUpToDegree(numerator, denominator, x, MaxReducedDegree)
+                : null;
+
+        /// <summary>
+        /// The integral of <paramref name="numerator"/> over <paramref name="denominator"/>,
+        /// held to <see cref="MaxDegree"/> unless asked otherwise: below a substitution a
+        /// denominator past it -- <c>1/(cos(x) + cos(3x))^5</c> under the tangent's half angle --
+        /// was reduced past the budget where it used to be declined at once.
+        /// </summary>
         internal static Entity? Integrate(Entity numerator, Entity denominator, Variable x)
+            => IntegrateUpToDegree(numerator, denominator, x, MaxDegree);
+
+        private static Entity? IntegrateUpToDegree(Entity numerator, Entity denominator, Variable x, int degree)
         {
-            if (!PolynomialFactoring.TryGetRationalCoefficients(numerator, x, 1, 0, MaxDegree, out var aboveCoefficients)
-                || !PolynomialFactoring.TryGetRationalCoefficients(denominator, x, 1, 1, MaxDegree, out var belowCoefficients))
+            if (!PolynomialFactoring.TryGetRationalCoefficients(numerator, x, 1, 0, degree, out var aboveCoefficients)
+                || !PolynomialFactoring.TryGetRationalCoefficients(denominator, x, 1, 1, degree, out var belowCoefficients))
                 return null;
             var above = RationalPolynomial.Create(aboveCoefficients);
             var below = RationalPolynomial.Create(belowCoefficients);
@@ -87,7 +115,7 @@ internal static class RothsteinTrager
             var answer = rationalPart;
             if (!logAbove.IsZero)
             {
-                if (LogarithmicPart(logAbove, logBelow, x) is not { } logarithmic)
+                if (logBelow.Degree > MaxDegree || LogarithmicPart(logAbove, logBelow, x) is not { } logarithmic)
                     return null;
                 answer = answer + logarithmic;
             }
 
@@ -0,0 +1,41 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using System;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// A rational function with numbers for coefficients over a denominator past the twelfth
+    /// degree, by Hermite's reduction: Rubi's 1.3.1 derivatives over powers of
+    /// <c>x^4 + x + 3</c>, written as sums of quotients over different powers of it.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AHighDegreeRationalFunctionIntegralTest
+    {
+        [Theory]
+        [InlineData("3*(-47+228*x+120*x^2+19*x^3)/(3+x+x^4)^4+(42-320*x-75*x^2-8*x^3)/(3+x+x^4)^3+30*x/(3+x+x^4)^2")]
+        [InlineData("(-3+10*x+4*x^3-30*x^5)/(3+x+x^4)^3-3*(1+4*x^3)*(2-3*x+5*x^2+x^4-5*x^6)/(3+x+x^4)^4")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            var derivative = integral.Substitute("C", 0).Differentiate("x");
+            foreach (var at in new[] { -2.3, -0.7, 0.4, 1.9 })
+            {
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                var want = integrand.ToEntity().Substitute("x", at).EvalNumerical();
+                var difference = Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart);
+                var scale = Math.Max(1.0, Math.Abs((double)want.RealPart));
+                Assert.True(difference / scale < 1e-8, $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+        }
+    }
+}