Name a logarithm of nested powers of one linear as a multiple of the linear's logarithm

BREAKING-CHANGES.md

@@ -565,6 +565,22 @@ the working digits, or `MathS.Settings.PrecisionErrorZeroRange` where a caller h
 still that integer, so `e^(-123.456)` is 0 as it was.
 [#1338](https://github.com/asc-community/AngouriMath/issues/1338).
 
+### A logarithm of nested powers of one linear is named as a multiple of the linear's logarithm
+
+**Answers where there were none, and shorter ones.** Rubi's 3.3 `(g + h x)^2 (a + b ln(c (d (e + f x)^p)^q))^3` was declined
+after eleven seconds and twelve gigabytes. The logarithm's derivative is `p q f/(e + f x)` on any branch, so it is
+`p q ln(e + f x) + K`, with `K` constant wherever it is continuous. Named so, the integrand is a polynomial times a power
+of `ln(e + f x)`, and the answer writes the logarithm back as it was. The square and the cube beside a linear come out
+shorter. The first power beside a quadratic comes out a little longer
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(g+h*x)^2*(a+b*ln(c*(d*(k+f*x)^p)^q))^3".ToEntity().Integrate("x")` | `integral(...)` | 1,660 characters, where before this change master declined it after 11 s and 12 GB |
+| `"(g+h*x)^2*(a+b*ln(c*(d*(k+f*x)^p)^q))^2".ToEntity().Integrate("x")` | `integral(...)` | 1,051 characters, where before this change master gave 2,048 |
+| `"(g+h*x)*(a+b*ln(c*(d*(k+f*x)^p)^q))^3".ToEntity().Integrate("x")` | `integral(...)` | 702 characters, where before this change master gave 2,569 |
+| `"(g+h*x)^2*(a+b*ln(c*(d*(k+f*x)^p)^q))".ToEntity().Integrate("x")` | `integral(...)` | 631 characters, where before this change master gave 527 |
+
 ### A power of a power of a logarithm over x is integrated below the question asked too
 
 **Answers where there were none.** Rubi's 3.5 `(ln(a x^n)^2)^p/x` was declined. Under `u = ln(a x^n)` it is

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

@@ -30090,6 +30090,69 @@ Entity Named(Entity value)
             return Functions.PartialFractions.DerivativeHoldsAtSampledPoints(answer, expr, x) ? answer : null;
         }
 
+        /// <summary>
+        /// A logarithm of constants and nested powers of one linear <c>L</c>, <c>ln(c (d L^p)^q)</c>, is
+        /// <c>p q ln(L) + K</c> with <c>K</c> constant wherever it is continuous, since its derivative is
+        /// <c>p q L'/L</c> by the chain rule on any branch: named so, Rubi's 3.3
+        /// <c>(g + h x)^2 (a + b ln(c (d (e + f x)^p)^q))^3</c> is a polynomial times a power of
+        /// <c>ln(e + f x)</c>, answered in a quarter of a second, where as written it was declined after
+        /// eleven and twelve gigabytes. The answer is the generic case's, as
+        /// <see cref="SolveByNamingALogarithmLinearInTheVariable"/>'s is. Only a nested power, which the
+        /// rules for <c>ln(c L^n)</c> do not read. Asked as a question of its own.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </summary>
+        internal static Entity? SolveByNamingALogarithmOfNestedPowersOfALinear(Entity expr, Entity.Variable x, bool integrateByParts)
+        {
+            Logf? named = null;
+            Entity? linear = null;
+            Entity? times = null;
+            foreach (var node in expr.Nodes)
+            {
+                if (node is not Logf(var logBase, var argument) log || logBase != MathS.e || !argument.ContainsNode(x)
+                    || !argument.Nodes.Any(inner => inner is Powf(var raised, _) && raised.ContainsNode(x) && raised.Nodes.Any(deeper => deeper != raised && deeper is Powf(var b, _) && b.ContainsNode(x))))
+                    continue;
+                Entity? found = null;
+                Entity? Exponent(Entity part)
+                {
+                    if (!part.ContainsNode(x))
+                        return Number.Integer.Zero;
+                    if (TreeAnalyzer.TryGetPolyLinear(part, x, out var slope, out _) && !TreeAnalyzer.IsZero(slope))
+                    {
+                        if (found is null)
+                            found = part;
+                        return found == part ? Number.Integer.One : null;
+                    }
+                    switch (part)
+                    {
+                        case Mulf(var left, var right):
+                            return Exponent(left) is { } l && Exponent(right) is { } r ? l + r : null;
+                        case Divf(var above, var below):
+                            return Exponent(above) is { } a && Exponent(below) is { } b ? a - b : null;
+                        case Powf(var @base, var power) when !power.ContainsNode(x):
+                            return Exponent(@base) is { } e ? power * e : null;
+                        default:
+                            return null;
+                    }
+                }
+                if (Exponent(argument) is not { } exponent || found is null)
+                    continue;
+                (named, linear, times) = (log, found, exponent.InnerSimplified);
+                break;
+            }
+            if (named is null || linear is null || times is null || TreeAnalyzer.IsZero(times))
+                return null;
+            var constant = Variable.CreateUnique(expr, "K");
+            var inTheLinear = times * MathS.Ln(linear) + constant;
+            var rewritten = expr.Substitute(named, inTheLinear);
+            if (rewritten == expr || rewritten.ContainsNode(named))
+                return null;
+            if (Integration.ComputeAsAQuestionOfItsOwn(rewritten, x, integrateByParts) is not { } answer)
+                return null;
+            // The sum as it was written stands for the logarithm again, where term by term it would be
+            // `p q ln(L) + (ln(c (d L^p)^q) - p q ln(L))`; K anywhere else is the difference.
+            return answer.Replace(node => node == inTheLinear ? named : node).Substitute(constant, named - times * MathS.Ln(linear));
+        }
+
         /// <summary>
         /// The sign of a real-valued function of <paramref name="x"/> among the factors of the
         /// integrand is constant on every interval between its zeros, and goes in front of the

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

@@ -695,6 +695,9 @@ internal static bool ASumOverRootsWouldAnswer
             // And a squared sine beside the squared cosine of its argument, the one they add up to:
             // `e^(cos(x)^2 + sin(x)^2)` was declined, written with an x its value does not hold.
             if ((answer = IndefiniteIntegralSolver.SolveByFoldingAPythagoreanPair(expr, x, integrateByParts)) is { }) return answer;
+            // A logarithm of nested powers of one linear, `ln(c (d (e + f x)^p)^q)`, named as `p q ln(e + f x)`
+            // plus a constant, before the rules that declined its cube beside a quadratic after twelve gigabytes.
+            if ((answer = IndefiniteIntegralSolver.SolveByNamingALogarithmOfNestedPowersOfALinear(expr, x, integrateByParts)) is { }) return answer;
             // An arctangent over a power of d (1 +- i c x), under x = tan(u)/c, where 1 + i tan(u) is
             // e^(i u)/cos(u): the rules below ran out of time on `(a + b atan(c x))^2/(d + i c d x)^2`.
             if ((answer = IndefiniteIntegralSolver.SolveAnArctangentOverAPowerOfOnePlusICXUnderTheTangent(expr, x, integrateByParts)) is { }) return answer;

Sources/Tests/UnitTests/Calculus/ALogarithmOfNestedPowersOfALinearIntegralTest.cs

@@ -0,0 +1,56 @@
+//
+// 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 System.Threading;
+using System.Threading.Tasks;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// A logarithm of nested powers of one linear, <c>ln(c (d (e + f x)^p)^q)</c>, named as
+    /// <c>p q ln(e + f x) + K</c>, K constant where it is continuous: Rubi's 3.3
+    /// <c>(g + h x)^2 (a + b ln(c (d (e + f x)^p)^q))^3</c> was declined after eleven seconds and twelve
+    /// gigabytes, where in the linear's logarithm it is a polynomial times a power of it.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ALogarithmOfNestedPowersOfALinearIntegralTest
+    {
+        [Theory]
+        // Rubi 3.3:620, with k for Rubi's e.
+        [InlineData("(g + h*x)^2*(a + b*ln(c*(d*(k + f*x)^p)^q))^3")]
+        [InlineData("(g + h*x)*(a + b*ln(c*(d*(k + f*x)^p)^q))^3")]
+        [InlineData("(g + h*x)^2*(a + b*ln(c*(d*(k + f*x)^p)^q))^2")]
+        public void IsAnsweredWithTheLogarithmAsWritten(string integrand)
+        {
+            Entity? integral = null;
+            // A thread of its own, so that the guard is not spent waiting behind other tests' work.
+            var task = Task.Factory.StartNew(() => integral = integrand.ToEntity().Integrate("x"),
+                CancellationToken.None, TaskCreationOptions.LongRunning, TaskScheduler.Default);
+            Assert.True(task.Wait(LimitTermination.Guard), $"{integrand} did not finish within {LimitTermination.Guard}");
+            var text = integral!.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.Contains("ln(c * (d * (k + f * x) ^ p) ^ q)", text);
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.4).Substitute("d", 0.9)
+                .Substitute("f", 0.8).Substitute("g", 1.7).Substitute("h", 0.6).Substitute("k", 0.2).Substitute("p", 1.5).Substitute("q", 0.75);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            foreach (var at in new[] { 0.3, 0.9, 1.4, 2.1, 3.3 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                Assert.False(got.IsNaN, $"d/dx of the antiderivative of {integrand} has no value at x = {at}, where the integrand is {want}");
+                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}");
+            }
+        }
+    }
+}