Take a linear factor common to a sum below the bar out of it

BREAKING-CHANGES.md

@@ -566,6 +566,23 @@ split. It is written `b (s + r)(s^2 - r s + r^2)` now, with `r = (a/b)^(1/3)`, w
 | `"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 linear factor common to a sum below the bar is taken out of it
+
+**Answers where there were none.** `1/(a x + b x/log(c x^n))` was left unevaluated. With x taken out of the sum below the
+bar it is `1/(x (a + b/log(c x^n)))`, a function of the logarithm beside its differential, which is answered. The same
+holds for the higher powers of the logarithm and for Rubi's 3.5:37
+([#718](https://github.com/asc-community/AngouriMath/issues/718)). `tanh(a + 2 log(x))^2/x^3`, which 2.5.0 left
+unevaluated, is answered by this route as well.
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"1/(a*x+b*x/ln(c*x^n))".ToEntity().Integrate("x")` | `integral(...)` | 72 characters |
+| `"1/(a*x+b*x/ln(c*x^n)^2)".ToEntity().Integrate("x")` | `integral(...)` | 503 characters |
+| `"1/(a*x+b*x/ln(c*x^n)^3)".ToEntity().Integrate("x")` | `integral(...)` | 514 characters |
+| `"1/(a*x+b*x/ln(c*x^n)^4)".ToEntity().Integrate("x")` | `integral(...)` | 337 characters |
+| `"(a*x+2*b*n*ln(c*x^n))/(a*x^2+b*x*ln(c*x^n)^2)".ToEntity().Integrate("x")` | `integral(...)` | 37 characters |
+| `"tanh(a+2*ln(x))^2/x^3".ToEntity().Integrate("x")` | `integral(...)` | 7,796 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

@@ -29334,6 +29334,37 @@ static bool IsZero(Entity value) => value.InnerSimplified.Evaled is Number.Compl
             return written == expr ? null : Integration.ComputeAsTheSameQuestion(written, x, integrateByParts);
         }
 
+        /// <summary>
+        /// A quotient whose denominator is a sum every term of which carries the same power of one
+        /// linear form in <paramref name="x"/>, with that power taken out below the bar:
+        /// <c>1/(a x + b x/ln(c x^n))</c> is <c>1/(x (a + b/ln(c x^n)))</c>, a function of the
+        /// logarithm times its differential.
+        /// </summary>
+        /// <remarks>
+        /// Rubi's 3.5 <c>1/(a x + b x/log(c x^n)^k)</c> were left unevaluated, and the same with
+        /// <c>x</c> taken out is answered in a tenth of a second.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </remarks>
+        internal static Entity? SolveByGatheringACommonLinearFactorOfASumBelowTheBar(Entity expr, Entity.Variable x, bool integrateByParts)
+        {
+            static bool IsASum(Entity e) => e is Sumf or Minusf;
+            Entity above, below;
+            var power = 1;
+            if (expr is Divf(var n, Powf(var d, Number.Integer k)) && IsASum(d) && k.EInteger.Sign > 0 && k.EInteger.CanFitInInt32())
+                (above, below, power) = (n, d, k.EInteger.ToInt32Unchecked());
+            else if (expr is Divf(var n1, var d1) && IsASum(d1))
+                (above, below) = (n1, d1);
+            else if (expr is Powf(var d2, Number.Integer k2) && IsASum(d2) && k2.EInteger.Sign < 0 && k2.EInteger.CanFitInInt32())
+                (above, below, power) = (Number.Integer.One, d2, -k2.EInteger.ToInt32Unchecked());
+            else
+                return null;
+            if (TryTakeACommonLinearFactorOfASum(below, x) is not var (factored, inside))
+                return null;
+            var taken = factored * inside;
+            var written = above / (power == 1 ? taken : MathS.Pow(factored, Number.Integer.Create(power)) * MathS.Pow(inside, Number.Integer.Create(power)));
+            return Integration.ComputeAsTheSameQuestion(written, x, integrateByParts);
+        }
+
         /// <summary>
         /// The sum with the highest power of one linear form in <paramref name="x"/> that every
         /// term carries taken out of it, as that power and what is left; <see langword="null"/>

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

@@ -1223,6 +1223,11 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             if (expr is Entity.Sumf or Entity.Minusf
                 && (answer = IndefiniteIntegralSolver.SolveByGatheringACommonPolynomialFactorOfASum(expr, x, integrateByParts)) is { })
                 return answer;
+            // The same factor taken out of a sum below the bar: `1/(a x + b x/ln(c x^n))` is
+            // `1/(x (a + b/ln(c x^n)))`, a function of the logarithm beside its differential.
+            if (expr is Entity.Divf or Entity.Powf
+                && (answer = IndefiniteIntegralSolver.SolveByGatheringACommonLinearFactorOfASumBelowTheBar(expr, x, integrateByParts)) is { })
+                return answer;
             // A power of the secant that is not whole beside the cosine, written as the cosine's
             // with the constant that takes in front, which the substitutions below read.
             if ((answer = IndefiniteIntegralSolver.SolveByWritingAPowerOfTheSecantInTheCosine(expr, x, integrateByParts)) is { }) return answer;

Sources/Tests/UnitTests/Calculus/ACommonLinearFactorOfASumBelowTheBarIntegralTest.cs

@@ -0,0 +1,57 @@
+//
+// 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 power of a linear common to every term of a sum below the bar, taken out: Rubi's 3.5
+    /// <c>1/(a x + b x/log(c x^n)^k)</c>, which was left unevaluated.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ACommonLinearFactorOfASumBelowTheBarIntegralTest
+    {
+        private static void DifferentiatesBack(string integrand, params (string, double)[] pins)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+
+            var pinned = integral.Substitute("C", 0);
+            Entity original = integrand.ToEntity();
+            foreach (var (name, value) in pins)
+            {
+                pinned = pinned.Substitute(name, value);
+                original = original.Substitute(name, value);
+            }
+            var derivative = pinned.Differentiate("x");
+            // Where the logarithm is real and away from the zeros of the denominator.
+            foreach (var at in new[] { 0.4, 1.9, 2.7, 3.3 })
+            {
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                var want = original.Substitute("x", at).EvalNumerical();
+                var difference = Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart);
+                var scale = Math.Max(1.0, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart));
+                Assert.True(difference / scale < 1e-8,
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+        }
+
+        [Theory]
+        [InlineData("1/(a*x+b*x/ln(c*x^n))")]
+        [InlineData("1/(a*x+b*x/ln(c*x^n)^2)")]
+        [InlineData("1/(a*x+b*x/ln(c*x^n)^3)")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("a", 1.3), ("b", 0.7), ("c", 2.5), ("n", 2));
+            DifferentiatesBack(integrand, ("a", 0.6), ("b", -0.2), ("c", 1.5), ("n", 3));
+        }
+    }
+}