x over a cubic binomial beside the root of another is integrated where that is elementary

BREAKING-CHANGES.md

@@ -747,6 +747,21 @@ or of its square over the root. Rubi's 1.3.3 names those coefficients, and its r
 | `"(c - 2*d*x)/((c + d*x)*sqrt(c^3 + 4*d^3*x^3))".ToEntity().Integrate("x")` | `integral(...)` | `2 c/d` times an arctangent of `sqrt(3 c^3) (1 + 2 d x/c)/sqrt(c^3 + 4 d^3 x^3)` over `sqrt(3 c^3)`, or an inverse hyperbolic tangent, by the sign of `c^3` |
 | `"(1+x)/((x-2)*sqrt(1+x^3))".ToEntity().Integrate("x")` | `integral(...)` | `-(2/3) artanh((1 + x)^2/(3 sqrt(1 + x^3)))`, written as a logarithm |
 
+### `x` over a cubic binomial beside the root of another is integrated where that is elementary
+
+**Answers where there were none.** `x/((a + b x^3) sqrt(c + d x^3))` is elementary at `4 b c = a d`
+and at `8 b c + a d = 0`, Rubi's 1.1.3.4, as arctangents and inverse hyperbolic tangents of
+`sqrt(c + d x^3)` and of `(1 + q x)/sqrt(c + d x^3)` for `q = (d/c)^(1/3)`, and Welz's
+`x/((4 - x^3) sqrt(1 - x^3))` and the rest at those ratios were declined. They are answered in
+closed form now, by the sign of `c`
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x/((4 - x^3)*sqrt(1 - x^3))".ToEntity().Integrate("x")` | `integral(...)` | two arctangents and two inverse hyperbolic tangents, of `sqrt(1 - x^3)` and of `(1 - 2^(1/3) x)/sqrt(1 - x^3)` and its kind |
+| `"x/((4*c + d*x^3)*sqrt(c + d*x^3))".ToEntity().Integrate("x")` | `integral(...)` | the same in `sqrt(c + d x^3)` and `q = (d/c)^(1/3)`, by the sign of `c` |
+| `"x/((8 - d*x^3)*sqrt(1 + d*x^3))".ToEntity().Integrate("x")` | `integral(...)` | `(artanh((1 + d^(1/3) x)^2/(3 sqrt(1 + d x^3))) - artanh(sqrt(1 + d x^3)/3))/(18 d^(2/3)) - atan(sqrt(3) (1 + d^(1/3) x)/sqrt(1 + d x^3))/(6 sqrt(3) d^(2/3))`, written with logarithms |
+
 ### `x^2` over a three-quarter power of a quadratic binomial beside another is integrated where that is elementary
 
 **Answers where there were none.** `x^2/((A + B x^2)^(3/4) (C + D x^2))` at `B C - 2 A D = 0`, Rubi's

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

@@ -4920,6 +4920,105 @@ Entity OverOnePlus(Entity k)
             return holds ? answer : null;
         }
 
+        /// <summary>
+        /// <c>x/((a + b x^3) sqrt(c + d x^3))</c> at the two ratios where its integral is elementary,
+        /// Rubi's 1.1.3.4: <c>4 b c = a d</c> and <c>8 b c + a d = 0</c>, in closed form.
+        /// </summary>
+        /// <remarks>
+        /// <para>
+        /// With <c>q = (d/c)^(1/3)</c>, the real root, and <c>y = sqrt(c + d x^3)</c>, at
+        /// <c>4 b c = a d</c>, for <c>r = sqrt(c)</c> where <c>c</c> is positive and
+        /// <c>r = sqrt(-c)</c> where it is negative:
+        /// <code>
+        /// c &gt; 0:   q/(3 2^(2/3) b r) (artanh(y/r)/3 + atan(y/(sqrt(3) r))/sqrt(3)
+        ///                      - atan(sqrt(3) r (1 + 2^(1/3) q x)/y)/sqrt(3) - artanh(r (1 - 2^(1/3) q x)/y))
+        /// c &lt; 0:  -q/(3 2^(2/3) b r) (atan(y/r)/3 + artanh(y/(sqrt(3) r))/sqrt(3)
+        ///                      + artanh(sqrt(3) r (1 + 2^(1/3) q x)/y)/sqrt(3) + atan(r (1 - 2^(1/3) q x)/y))
+        /// </code>
+        /// and at <c>8 b c + a d = 0</c>, where the integrand is <c>-(d/b) x/((8 c - d x^3) y)</c>, that
+        /// times
+        /// <code>
+        /// c &gt; 0:   (artanh(r (1 + q x)^2/(3 y))/18 - artanh(y/(3 r))/18 - atan(sqrt(3) r (1 + q x)/y)/(6 sqrt(3)))/(r^3 q^2)
+        /// c &lt; 0:  -(atan(r (1 + q x)^2/(3 y))/18 + atan(y/(3 r))/18 - artanh(sqrt(3) r (1 + q x)/y)/(6 sqrt(3)))/(r^3 q^2)
+        /// </code>
+        /// Rubi splits the second into three integrals, a substitution's in <c>x^3</c>, a linear
+        /// over a linear and a quadratic over a quadratic, each elementary, and their sum is the
+        /// form above; asked as three, two come back piecewise in the symbols and their sum holds
+        /// every combination of the arms. Every condition is decided as a value, and the answer is
+        /// differentiated back at sampled points before it is returned.
+        /// Welz's <c>x/((4 - x^3) sqrt(1 - x^3))</c> and <c>x/((8 - d x^3) sqrt(1 + d x^3))</c> were
+        /// declined, with the rest of 1.1.3.4 at those ratios.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </para>
+        /// </remarks>
+        internal static Entity? SolveXOverACubicBinomialBesideTheRootOfAnother(Entity expr, Entity.Variable x, bool integrateByParts)
+        {
+            var (numerator, denominator) = Functions.SingleQuotient.Of(expr);
+            if (!TreeAnalyzer.TryGetPolynomial(numerator, x, out var above) || above.Count != 1
+                || !above.TryGetValue(EInteger.One, out var multiple) || multiple.ContainsNode(x))
+                return null;
+            Entity? radicand = null;
+            Entity below = Number.Integer.One;
+            foreach (var factor in Mulf.LinearChildren(denominator))
+                if (factor is Powf(var @base, Number.Rational half) && half.ERational.Equals(ERational.Create(1, 2)) && @base.ContainsNode(x))
+                {
+                    if (radicand is not null)
+                        return null;
+                    radicand = @base;
+                }
+                else
+                    below = below == Number.Integer.One ? factor : below * factor;
+            if (radicand is null || ACubicBinomial(radicand) is not var (c, d) || ACubicBinomial(below) is not var (a, b))
+                return null;
+            static bool Zero(Entity condition) => Functions.PartialFractions.IsZeroAsAValue(condition);
+            if (Zero(a) || Zero(b) || Zero(c) || Zero(d) || Zero(b * c - a * d))
+                return null;
+            var y = MathS.Sqrt(radicand);
+            var q = MathS.Pow(LowestOverTheSymbols(d / c), Number.Rational.Create(1, 3));
+            var cubeRootOfTwo = MathS.Pow(2, Number.Rational.Create(1, 3));
+            var sqrtOfThree = MathS.Sqrt(3);
+            Entity? answer = null;
+            if (Zero(4 * b * c - a * d))
+            {
+                var r = MathS.Sqrt(c);
+                var wherePositive = q / (3 * MathS.Sqr(cubeRootOfTwo) * b * r)
+                    * (MathS.Hyperbolic.Artanh(y / r) / 3 + MathS.Arctan(y / (sqrtOfThree * r)) / sqrtOfThree
+                       - MathS.Arctan(sqrtOfThree * r * (1 + cubeRootOfTwo * q * x) / y) / sqrtOfThree
+                       - MathS.Hyperbolic.Artanh(r * (1 - cubeRootOfTwo * q * x) / y));
+                var s = MathS.Sqrt(LowestOverTheSymbols(-c));
+                var whereNegative = -q / (3 * MathS.Sqr(cubeRootOfTwo) * b * s)
+                    * (MathS.Arctan(y / s) / 3 + MathS.Hyperbolic.Artanh(y / (sqrtOfThree * s)) / sqrtOfThree
+                       + MathS.Hyperbolic.Artanh(sqrtOfThree * s * (1 + cubeRootOfTwo * q * x) / y) / sqrtOfThree
+                       + MathS.Arctan(s * (1 - cubeRootOfTwo * q * x) / y));
+                answer = BySign(c, wherePositive, whereNegative);
+            }
+            else if (Zero(8 * b * c + a * d))
+            {
+                // The integrand is -(d/b) x/((8 c - d x^3) y).
+                var r = MathS.Sqrt(c);
+                var wherePositive = (MathS.Hyperbolic.Artanh(r * MathS.Sqr(1 + q * x) / (3 * y)) / 18 - MathS.Hyperbolic.Artanh(y / (3 * r)) / 18
+                    - MathS.Arctan(sqrtOfThree * r * (1 + q * x) / y) / (6 * sqrtOfThree)) / (MathS.Pow(r, 3) * MathS.Sqr(q));
+                var s = MathS.Sqrt(LowestOverTheSymbols(-c));
+                var whereNegative = -(MathS.Arctan(s * MathS.Sqr(1 + q * x) / (3 * y)) / 18 + MathS.Arctan(y / (3 * s)) / 18
+                    - MathS.Hyperbolic.Artanh(sqrtOfThree * s * (1 + q * x) / y) / (6 * sqrtOfThree)) / (MathS.Pow(s, 3) * MathS.Sqr(q));
+                answer = LowestOverTheSymbols(-d / b) * BySign(c, wherePositive, whereNegative);
+            }
+            if (answer is null)
+                return null;
+            answer = (multiple * answer).InnerSimplified;
+            bool holds;
+            using (MathS.Settings.DowncastingEnabled.Set(false))
+                holds = Functions.PartialFractions.HoldsAtSampledPoints(answer.Differentiate(x), expr, x);
+            return holds ? answer : null;
+
+            // `p + r x^3`, as `(p, r)`, each free of x.
+            (Entity, Entity)? ACubicBinomial(Entity binomial)
+                => TreeAnalyzer.TryGetPolynomial(binomial, x, out var read) && read.Count == 2
+                   && read.TryGetValue(EInteger.Zero, out var constant) && read.TryGetValue(EInteger.FromInt32(3), out var cubic)
+                   && !constant.ContainsNode(x) && !cubic.ContainsNode(x)
+                    ? (constant, cubic) : null;
+        }
+
         /// <summary>
         /// A root of a quadratic binomial beside another, <c>1/((A + B x^2)^(1/3) (C + D x^2))</c>
         /// with <c>B C + 3 A D = 0</c> or <c>B C - 9 A D = 0</c>, and

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

@@ -807,6 +807,9 @@ private static Entity Normalized(Entity expr, Entity.Variable x) =>
             // `(1 + x + sqrt(3))/((1 + x - sqrt(3)) sqrt(1 + x^3))`. Before the reduction over the
             // linear, which writes it as `1` and `2 sqrt(3)` over the linear, neither elementary.
             if ((answer = IndefiniteIntegralSolver.SolveAPseudoEllipticQuotientOverTheRootOfACubicBinomial(expr, x)) is { }) return answer;
+            // x over a cubic binomial beside the root of another, at the two ratios where that is
+            // elementary: Welz's `x/((4 - x^3) sqrt(1 - x^3))`.
+            if ((answer = IndefiniteIntegralSolver.SolveXOverACubicBinomialBesideTheRootOfAnother(expr, x, integrateByParts)) is { }) return answer;
             if ((answer = IndefiniteIntegralSolver.SolveByReducingThePolynomialOverALinearFactor(expr, x, integrateByParts)) is { }) return answer;
             if (expr is Entity.Sumf or Entity.Minusf
                 && (answer = IndefiniteIntegralSolver.SolveBySplittingSum(expr, x, integrateByParts)) is { })

Sources/Tests/UnitTests/Calculus/XOverACubicBinomialBesideTheRootOfAnotherIntegralTest.cs

@@ -0,0 +1,62 @@
+//
+// 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>
+    /// <c>x/((a + b x^3) sqrt(c + d x^3))</c> at the two ratios where its integral is elementary,
+    /// Rubi's 1.1.3.4: <c>4 b c = a d</c>, in closed form by the sign of <c>c</c>, and
+    /// <c>8 b c + a d = 0</c>, as three integrals.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    /// <remarks>
+    /// Checked by differentiating back with <c>c</c> pinned to either sign and <c>d = 0.7</c>, on
+    /// both sides of 0 wherever the integrand is real.
+    /// </remarks>
+    [Trait("Area", "Calculus")]
+    public sealed class XOverACubicBinomialBesideTheRootOfAnotherIntegralTest
+    {
+        private static readonly double[] Points = { -2.3, -1.4, -0.8, -0.3, 0.4, 0.9, 1.6, 2.4 };
+
+        [Theory]
+        // 4 b c = a d.
+        [InlineData("x/((4 - x^3)*sqrt(1 - x^3))", 1.3)]
+        [InlineData("x/((4 - d*x^3)*sqrt(-1 + d*x^3))", 1.3)]
+        [InlineData("x/((4*c + d*x^3)*sqrt(c + d*x^3))", 1.3)]
+        [InlineData("x/((4*c + d*x^3)*sqrt(c + d*x^3))", -1.3)]
+        // 8 b c + a d = 0.
+        [InlineData("x/((8 - d*x^3)*sqrt(1 + d*x^3))", 1.3)]
+        [InlineData("x/((8*c - d*x^3)*sqrt(c + d*x^3))", 1.3)]
+        [InlineData("x/((8*c - d*x^3)*sqrt(c + d*x^3))", -1.3)]
+        [InlineData("3*x/((8 + x^3)*sqrt(-1 + x^3))", 1.3)]
+        public void IsIntegrated(string integrand, double c)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Entity Pinned(Entity e) => e.Substitute("c", c).Substitute("d", 0.7);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in Points)
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                if (want.IsNaN || Math.Abs((double)want.ImaginaryPart) > 1e-12 * Math.Max(1, Math.Abs((double)want.RealPart)))
+                    continue;
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                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)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+            Assert.True(compared >= 2, $"only {compared} points were real for {integrand}");
+        }
+    }
+}