Split a rational function over a binomial of degree three or four beside other factors

BREAKING-CHANGES.md

@@ -614,6 +614,23 @@ carried the discriminant's being real in its condition, and held nowhere when it
 | `"piecewise(2 * x provided not x = 0, 3 * x ^ 2 provided i - 1 < 0)".ToEntity().DomainCondition` | `false` | `not x = 0` |
 | `"1/(sqrt(a + i*a*x)*(c + d*x)^(3/2)*(a - i*a*x))".ToEntity().Integrate("x")` | `integral(...)`; on the unreleased master, an answer with no value at any point | 3,267 characters, defined |
 
+### A rational function over a binomial of degree three or four beside other factors is split over them
+
+**Answers where there were none.** Rubi's 4.3.7 `1/(a + b tan(c + d x)^3)` and its square, and 6.1.7
+`sech(c + d x)/(a + b sqrt(sinh(c + d x)))` and its square, were declined. Under the tangent and under the root of the
+sine they are rational functions over `(a + b u^3) (1 + u^2)` and `(a + b u) (1 + u^4)`, and the split over the
+factors as written read linear and quadratic ones only. It now reads one binomial of degree three or four, written
+once, beside factors of degree one or two, and the rules for a polynomial over a binomial answer each piece; not
+beside a power of the variable alone, which those rules read term by term, nor beside a second such binomial
+([#233](https://github.com/asc-community/AngouriMath/issues/233)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"1/(a+b*tan(c+d*x)^3)".ToEntity().Integrate("x")` | `integral(...)` | 1,853 characters |
+| `"1/(a+b*tan(c+d*x)^3)^2".ToEntity().Integrate("x")` | `integral(...)` | 3,026 characters |
+| `"sech(c+d*x)/(a+b*sqrt(sinh(c+d*x)))".ToEntity().Integrate("x")` | `integral(...)` | 3,325 characters |
+| `"sech(c+d*x)/(a+b*sqrt(sinh(c+d*x)))^2".ToEntity().Integrate("x")` | `integral(...)` | 4,614 characters |
+
 ### A rational function over the quadratic of an inverse tangent beside it is split
 
 **Answers where there were none.** `x^4 atan(a x)/(c + a^2 c x^2)` ran out of time, and so did three more of Rubi's 5.3.4.

Sources/AngouriMath/Functions/Algebra/Polynomials/PartialFractions.cs

@@ -438,6 +438,8 @@ internal static bool TrySplitOverWrittenFactors(
             // coefficients free of x; a factor free of x is a constant and comes out in front.
             var factors = new List<(Entity Factor, int Degree)>();
             Entity constant = Integer.One;
+            var binomialsPastTheSecondDegree = 0;
+            var aPowerOfXAlone = false;
             foreach (var part in Mulf.LinearChildren(denominator))
             {
                 if (!part.ContainsNode(x))
@@ -464,8 +466,16 @@ internal static bool TrySplitOverWrittenFactors(
                 if (!TreeAnalyzer.TryGetPolynomial(toRead, x, out var read) || read.Count == 0)
                     return false;
                 var degree = read.Keys.Max()!;
-                if (!degree.CanFitInInt32() || degree.ToInt32Unchecked() is not (1 or 2))
+                // And a binomial of degree three or four written once, `a + b x^3`: the system below
+                // reads any degree, and the rules for a polynomial over a binomial answer each
+                // piece -- `1/((a + b u^3) (1 + u^2))`, which the tangent makes of Rubi's
+                // `1/(a + b tan(c + d x)^3)`, was declined for want of a split.
+                var aBinomialOnce = read.Count == 2 && multiplicity == 1 && degree.CanFitInInt32() && degree.ToInt32Unchecked() is 3 or 4;
+                if (!degree.CanFitInInt32() || degree.ToInt32Unchecked() is not (1 or 2) && !aBinomialOnce)
                     return false;
+                if (aBinomialOnce)
+                    binomialsPastTheSecondDegree++;
+                aPowerOfXAlone |= read.Count == 1;
                 // A repeated quadratic has a rule to land on only with a symbol somewhere,
                 // the split by residues below; over the rationals the system would hand it a
                 // cubic over the square, which no rule reads.
@@ -476,6 +486,15 @@ internal static bool TrySplitOverWrittenFactors(
                         return false;
                 factors.Add((part.InnerSimplified, degree.ToInt32Unchecked() * multiplicity));
             }
+            // Not beside a power of x alone, which is the rules' for `x^m` over a binomial, term by
+            // term: the cube root's substitution makes Rubi's 1.1.3.4 `(a + b x^3)^(1/3)/(x^5 (c + d x^3))`
+            // a polynomial over `u^3 (c + (a d - b c) u^3)` with `a^9 c^36 d^9` in it, and split over
+            // the two, it came back in 12,377 characters where those rules write 1,964. Nor beside a
+            // second such binomial: `x^7 (a + b x^3)^(1/3)/(c + d x^3)` is one over `(b u^3 - 1)` and
+            // `(c + (a d - b c) u^3)`, whose system with symbols in it came back with `a^17 b^16 c^43`
+            // and a wrong answer, where the rules for the cube root answer it in 2,861 characters.
+            if (binomialsPastTheSecondDegree > 1 || binomialsPastTheSecondDegree == 1 && aPowerOfXAlone)
+                return false;
             // Two linear factors with one root are one factor: `(a + b x)(a + x b)^2` is
             // written so by the rules that make a factor monic and gather its powers, and
             // as two distinct factors the decomposition has no answer -- its coefficients

Sources/Tests/UnitTests/Calculus/ARationalFunctionOverAWrittenBinomialOfDegreeThreeOrFourIntegralTest.cs

@@ -0,0 +1,86 @@
+//
+// 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 symbols in it over a product written with a binomial of degree three
+    /// or four in it, split over the factors as written by undetermined coefficients: each piece is a
+    /// polynomial over a binomial, or over a linear or a quadratic, which the rules for those answer.
+    /// The split read only linear and quadratic factors, and `1/((a + b u^3) (1 + u^2))`, which the
+    /// tangent makes of Rubi's 4.3.7 `1/(a + b tan(c + d x)^3)`, was declined.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ARationalFunctionOverAWrittenBinomialOfDegreeThreeOrFourIntegralTest
+    {
+        [Theory]
+        [InlineData("1/((a+b*x^3)*(1+x^2))")]
+        [InlineData("1/((a+b*x^4)*(c+x^2))")]
+        [InlineData("1/(a+b*tan(c+d*x)^3)")]
+        [InlineData("1/(a+b*tan(c+d*x)^3)^2")]
+        public void DifferentiatesBack(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.4).Substitute("d", 0.9);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -1.3, -0.6, -0.2, 0.3, 0.8, 1.4 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                if (want.IsNaN || got.IsNaN)
+                    continue;
+                compared++;
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart)
+                            < 1e-9 * Math.Max(1, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+            Assert.True(compared >= 5, $"only {compared} points could be compared for {integrand}");
+        }
+
+        /// <summary>
+        /// Not split beside a power of the variable alone, nor beside a second such binomial: the cube
+        /// root's substitution makes Rubi's 1.1.3.4 rows rational functions over <c>u^3 (c + k u^3)</c>
+        /// and over <c>(b u^3 - 1)(c + k u^3)</c>, and split, they came back in 12,377 and 27,590
+        /// characters with <c>c^36</c> and <c>c^43</c> in them, the second wrong, where the rules for
+        /// the cube root write 1,964 and 2,861.
+        /// </summary>
+        [Theory]
+        [InlineData("(a+b*x^3)^(1/3)/(x^5*(c+d*x^3))")]
+        [InlineData("x^7*(a+b*x^3)^(1/3)/(c+d*x^3)")]
+        public void IsNotSplitBesideAPowerOfXOrASecondBinomial(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 5000, $"the antiderivative of {integrand} is {text.Length} characters");
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.4).Substitute("d", 0.9);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { 0.3, 0.6, 0.9, 1.2, 1.5, 1.8 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                if (want.IsNaN || got.IsNaN)
+                    continue;
+                compared++;
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart)
+                            < 1e-9 * Math.Max(1, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+            Assert.True(compared >= 5, $"only {compared} points could be compared for {integrand}");
+        }
+    }
+}