Integrate a power of a power of a logarithm over x below the question asked

BREAKING-CHANGES.md

@@ -398,6 +398,20 @@ 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 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
+`(u^2)^p/n`, a symbolic power of a power of the variable, and the rule for powers of one linear answered that only
+where it was the question asked; below it, the rule now reads powers of the variable itself, with no sum in it. Its integral is
+`u (u^2)^p/(2 p + 1)` on both sides of `u = 0`, written as `(u^2)^p/u^(2 p)`, constant on each side, times
+`u^(2 p + 1)/(2 p + 1)` ([#233](https://github.com/asc-community/AngouriMath/issues/233)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(ln(x)^2)^p/x".ToEntity().Integrate("x")` | `integral(...)` | 79 characters |
+| `"(ln(a*x^n)^2)^p/x".ToEntity().Integrate("x")` | `integral(...)` | 107 characters |
+| `"(ln(a*x)^3)^p/x".ToEntity().Integrate("x")` | `integral(...)` | 91 characters |
+
 ### A polynomial over a power of the linear of an artanh is written in powers of the linear
 
 **Answers where there were none.** `x (a + b atanh(c x))^2/(d + c d x)^3`, Rubi's 7.3.4, ran out of time. In powers of

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

@@ -26501,7 +26501,10 @@ Entity ToThe(EInteger k)
         /// </remarks>
         internal static Entity? SolvePowersOfOneLinearAsOnePower(Entity expr, Entity.Variable x, bool integrateByParts)
         {
-            if (!Integration.AnsweringTheQuestionAsked)
+            // Below the top only powers of x itself, with no sum in x to read: `(u^2)^p`, which
+            // `u = ln(x)` makes of Rubi's 3.5 `(ln(x)^2)^p/x`, is `u (u^2)^p/(2 p + 1)` and was
+            // declined there.
+            if (!Integration.AnsweringTheQuestionAsked && expr.Nodes.Any(node => node is Sumf or Minusf && node.ContainsNode(x)))
                 return null;
             // Without a leading `1 *`, which is another spelling to the rules that read the result.
             static Entity Times(Entity a, Entity b) => a == Number.Integer.One ? b : a * b;

Sources/Tests/UnitTests/Calculus/APowerOfASquaredLogarithmOverXIntegralTest.cs

@@ -0,0 +1,50 @@
+//
+// 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 symbolic power of a power of the variable, asked below the question: <c>u = ln(x)</c> makes
+    /// Rubi's 3.5 <c>(ln(x)^2)^p/x</c> the question <c>(u^2)^p</c>, which is <c>u (u^2)^p/(2 p + 1)</c>
+    /// on every branch and was answered only where it was the question asked.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class APowerOfASquaredLogarithmOverXIntegralTest
+    {
+        [Theory]
+        [InlineData("(ln(x)^2)^p/x")]
+        [InlineData("(ln(a*x^n)^2)^p/x")]
+        [InlineData("(ln(a*x)^3)^p/x")]
+        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("n", 2.0).Substitute("p", 0.35);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            // Both sides of ln = 0, where (ln^2)^p is real and ln^(2p) is not.
+            foreach (var at in new[] { 0.2, 0.5, 0.8, 1.4, 2.6, 4.1 })
+            {
+                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}");
+        }
+    }
+}