A sine's binomial cube below the bar is split over the cube's two factors

BREAKING-CHANGES.md

@@ -249,6 +249,24 @@ roots, `1 - x^2`, is taken as its two ([#718](https://github.com/asc-community/A
 | `"1/((1 - x^2)*(a + b*x^3))".ToEntity().Integrate("x")` | `integral(...)` | 1,600 characters |
 | `"1/(x*(1 - x^2)*(a + b*x^3))".ToEntity().Integrate("x")` | `integral(...)` | 1,637 characters |
 
+### A sine's binomial cube below the bar is split over the cube's two factors
+
+**Answers where there were none.** `csc(c + d x)/(a + b sin(c + d x)^3)` was declined, and so were the second to fifth
+powers of the cosecant and the second and fourth powers of the secant over the same block (Rubi's 4.1.7). With `a`
+and `b` symbols the block has no rational factor, so the split over a symbolic quadratic in the sine had nothing to
+split. It is written `b (s + r)(s^2 - r s + r^2)` now, with `r = (a/b)^(1/3)`, which that split answers
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"csc(c + d*x)/(a + b*sin(c + d*x)^3)".ToEntity().Integrate("x")` | `integral(...)` | 5,058 characters |
+| `"csc(c + d*x)^2/(a + b*sin(c + d*x)^3)".ToEntity().Integrate("x")` | `integral(...)` | 4,866 characters |
+| `"csc(c + d*x)^3/(a + b*sin(c + d*x)^3)".ToEntity().Integrate("x")` | `integral(...)` | 5,382 characters |
+| `"csc(c + d*x)^4/(a + b*sin(c + d*x)^3)".ToEntity().Integrate("x")` | `integral(...)` | 5,426 characters |
+| `"csc(c + d*x)^5/(a + b*sin(c + d*x)^3)".ToEntity().Integrate("x")` | `integral(...)` | 5,958 characters |
+| `"sec(c + d*x)^2/(a + b*sin(c + d*x)^3)".ToEntity().Integrate("x")` | `integral(...)` | 7,010 characters |
+| `"sec(c + d*x)^4/(a + b*sin(c + d*x)^3)".ToEntity().Integrate("x")` | `integral(...)` | 11,646 characters |
+
 ### A symbol shared by a numerator sum's coefficients is taken out before the partial fractions
 
 **Answers where there were none.** `(b d + 2c d x)^9/(a + b x + c x^2)^3` ran past a minute, where

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

@@ -23744,10 +23744,59 @@ static bool IsASumOfPowersOfX(Entity sum, Entity.Variable x)
                     return null;
                 if (OverASymbolicQuadraticInTheFunction(inS, s, w, function == sine, argument, rate, x, integrateByParts) is { } answer)
                     return answer;
+                if (WithASymbolicBinomialCubeFactored(inS, s, w) is { } cube
+                    && OverASymbolicQuadraticInTheFunction(cube.Factored, s, w, function == sine, argument, rate, x, integrateByParts) is { } inR)
+                    return inR.Substitute(cube.R, cube.Root);
             }
             return null;
         }
 
+        /// <summary>
+        /// <paramref name="inS"/> with one factor <c>a + b s^3</c> below the bar, a symbol in it,
+        /// written <c>b (s + r)(s^2 - r s + r^2)</c> for a fresh <c>r</c> whose cube is <c>a/b</c>:
+        /// a linear and a quadratic with a symbol in it, which the rule above splits over. Null
+        /// where there is no such factor or more than one.
+        /// </summary>
+        /// <remarks>
+        /// Rubi's 4.1.7: <c>csc(x)/(a + b sin(x)^3)</c> and its even powers of the cosecant and
+        /// the secant were declined. Under the half-angle the cube is a sextic in <c>t</c> with
+        /// symbols in every coefficient. The identity holds for any cube root of <c>a/b</c>.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </remarks>
+        private static (Entity Factored, Variable R, Entity Root)? WithASymbolicBinomialCubeFactored(Entity inS, Variable s, Variable w)
+        {
+            var combined = Functions.SingleQuotient.Combine(inS).InnerSimplified;
+            if (combined is Providedf(var bare, _))
+                combined = bare;
+            var (numerator, denominator) = Functions.SingleQuotient.Of(combined);
+            Entity? cube = null;
+            Entity rest = Number.Integer.One;
+            Entity free = Number.Integer.Zero, leading = Number.Integer.Zero;
+            foreach (var factor in Mulf.LinearChildren(denominator))
+            {
+                if (factor is not (Sumf or Minusf) || factor.ContainsNode(w)
+                    || !TreeAnalyzer.TryGetPolynomial(factor, s, out var read)
+                    || read.Count != 2 || !read.ContainsKey(EInteger.Zero) || !read.ContainsKey(EInteger.FromInt32(3))
+                    || read.Values.Any(coefficient => coefficient.ContainsNode(s))
+                    || !factor.Vars.Any(v => v != s))
+                {
+                    rest = rest == Number.Integer.One ? factor : rest * factor;
+                    continue;
+                }
+                if (cube is not null)
+                    return null;
+                cube = factor;
+                free = read[EInteger.Zero];
+                leading = read[EInteger.FromInt32(3)];
+            }
+            if (cube is null || Functions.PartialFractions.IsZeroAsAValue(free) || Functions.PartialFractions.IsZeroAsAValue(leading))
+                return null;
+            var r = Variable.CreateUnique(inS, "r_cube");
+            var factored = leading * (s + r) * (MathS.Pow(s, 2) - r * s + MathS.Pow(r, 2));
+            var below = rest == Number.Integer.One ? factored : rest * factored;
+            return (numerator / below, r, MathS.Pow(free / leading, Number.Rational.Create(1, 3)));
+        }
+
         /// <summary>
         /// The rule above in <paramref name="s"/>, for the function that <paramref name="s"/>
         /// stands for and <paramref name="w"/> its complement: null where the integrand is

Sources/Tests/UnitTests/Calculus/ABinomialCubeInTheSineIntegralTest.cs

@@ -0,0 +1,44 @@
+//
+// 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 of the sine over <c>a + b sin^3</c>, the cube written
+    /// <c>b (s + r)(s^2 - r s + r^2)</c> with <c>r^3 = a/b</c> and split over its factors. Rubi's 4.1.7.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ABinomialCubeInTheSineIntegralTest
+    {
+        [Theory]
+        [InlineData("csc(c + d*x)/(a + b*sin(c + d*x)^3)")]
+        [InlineData("csc(c + d*x)^2/(a + b*sin(c + d*x)^3)")]
+        [InlineData("csc(c + d*x)^3/(a + b*sin(c + d*x)^3)")]
+        [InlineData("sec(c + d*x)^2/(a + b*sin(c + d*x)^3)")]
+        public void DifferentiatesBack(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Entity Pinned(Entity e) => e.Substitute("a", 2).Substitute("b", 3).Substitute("c", "1/4".ToEntity()).Substitute("d", "3/2".ToEntity());
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            foreach (var at in new[] { "0.35", "0.6", "1.1", "1.4" })
+            {
+                var want = original.Substitute("x", at.ToEntity()).EvalNumerical();
+                var got = derivative.Substitute("x", at.ToEntity()).EvalNumerical();
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart)
+                            < 1e-9 * Math.Max(1, Math.Abs((double)want.RealPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+        }
+    }
+}