Integrate a rational function times a logarithm of powers of linears by parts

BREAKING-CHANGES.md

@@ -396,6 +396,22 @@ in Rubi's 1.1.3.4 come out shorter as well, and those that took two seconds take
 | `"1/(x^8*(a + b*x^3)*(c + d*x^3))".ToEntity().Integrate("x")` | `integral(...)` | 1,217 characters |
 | `"1/(x^6*(a + b*x^4)*(c + d*x^4))".ToEntity().Integrate("x")` | `integral(...)` | 795 characters |
 
+### A rational function times a logarithm of nested powers of linears is integrated by parts
+
+**Answers where there were none.** `ln(e (f (a + b x)^p (c + d x)^q)^r)` was declined, and so were four more of Rubi's
+3.2.3 and 3.3. `integral R ln(G)` is now `S ln(G) - integral S G'/G` with `S = integral R` rational, and `G'/G` read off
+`G` as it is written -- a product's is the sum of its factors', a power's the exponent times its base's, a linear's
+`b/(a + b x)` -- which is exact by the chain rule on any branch, and rational
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"ln(k*(f*(a+b*x)^p*(c+d*x)^q)^r)".ToEntity().Integrate("x")` | `integral(...)` | 167 characters |
+| `"x*ln(k*(f*(a+b*x)^p*(c+d*x)^q)^r)".ToEntity().Integrate("x")` | `integral(...)` | 241 characters |
+| `"ln(k*(f*(a+b*x)^p*(c+d*x)^q)^r)/(a+b*x)^5".ToEntity().Integrate("x")` | `integral(...)` | 817 characters |
+| `"ln(k*(f*(a+b*x)^p*(c+d*x)^q)^r)/(g+h*x)^3".ToEntity().Integrate("x")` | `integral(...)` | 960 characters |
+| `"(a+b*ln(c*(d*(k+f*x)^p)^q))/(g+h*x)^2".ToEntity().Integrate("x")` | `integral(...)` | 261 characters |
+
 ### A function of an inverse tangent times its derivative up to a constant factor is integrated by substitution
 
 **Answers where there were none.** `atan(c x/sqrt(a - c^2 x^2))^m/sqrt(d - c^2 d x^2/a)` was declined, and so were

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

@@ -29184,6 +29184,88 @@ bool HoldsItsValue(Entity factor)
                 => Functions.PartialFractions.HoldsAtSampledPoints(factor, factor.Substitute(x, Number.Rational.Create(3, 10)), x);
         }
 
+        /// <summary>
+        /// A rational function <c>R</c> times the logarithm of <c>G</c>, a product of powers of linears
+        /// nested in powers and constants, by parts: <c>integral R ln(G) = S ln(G) - integral S G'/G</c>
+        /// with <c>S = integral R</c> rational, and <c>G'/G</c> read off <c>G</c> as it is written --
+        /// a product's is the sum of its factors', a power's the exponent times its base's, a linear's
+        /// <c>b/(a + b x)</c> -- which is exact by the chain rule on any branch, and rational.
+        /// </summary>
+        /// <remarks>
+        /// Rubi's 3.2.3 <c>ln(e (f (a + b x)^p (c + d x)^q)^r)</c>, over <c>(a + b x)^5</c> and over
+        /// <c>(g + h x)^3</c>, were declined or ran out of time.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </remarks>
+        internal static Entity? SolveARationalFunctionTimesALogarithmOfPowersOfLinears(Entity expr, Entity.Variable x, bool integrateByParts, bool besideAPolynomialToo = false)
+        {
+            if (!Integration.AnsweringTheQuestionAsked)
+                return null;
+            var (numerator, denominator) = Functions.SingleQuotient.Of(expr);
+            Logf? logarithm = null;
+            Entity rest = Number.Integer.One;
+            foreach (var factor in Mulf.LinearChildren(numerator))
+            {
+                if (factor is Logf { Base: var lb } found && lb == MathS.e && found.Antilogarithm.ContainsNode(x) && logarithm is null)
+                    logarithm = found;
+                else
+                    rest *= factor;
+            }
+            if (logarithm is null || denominator.Nodes.Any(node => node is Logf))
+                return null;
+            var g = logarithm.Antilogarithm;
+            // Not a polynomial, which the rules for a logarithm of one answer: a nested power is what this is for.
+            if (TreeAnalyzer.TryGetPolynomial(g, x, out _))
+                return null;
+            Entity? Derivative(Entity node)
+            {
+                if (!node.ContainsNode(x))
+                    return Number.Integer.Zero;
+                if (node is Mulf or Divf)
+                {
+                    var (top, bottom) = Functions.SingleQuotient.Of(node);
+                    Entity sum = Number.Integer.Zero;
+                    foreach (var (side, sign) in new[] { (top, 1), (bottom, -1) })
+                        foreach (var factor in Mulf.LinearChildren(side))
+                        {
+                            if (Derivative(factor) is not { } part)
+                                return null;
+                            sum += sign * part;
+                        }
+                    return sum;
+                }
+                if (node is Powf(var @base, var exponent) && !exponent.ContainsNode(x))
+                    return Derivative(@base) is { } inner ? exponent * inner : null;
+                if (TreeAnalyzer.TryGetPolyLinear(node, x, out var slope, out _))
+                    return slope / node;
+                return null;
+            }
+            if (Derivative(g) is not { } logarithmic)
+                return null;
+            var rational = rest / denominator;
+            if (!TreeAnalyzer.TryGetPolynomial(Functions.SingleQuotient.Of(rational).Numerator, x, out var readAbove)
+                || !TreeAnalyzer.TryGetPolynomial(Functions.SingleQuotient.Of(rational).Denominator, x, out _))
+                return null;
+            // A polynomial of degree one or more beside the logarithm is left to by parts as it was, and
+            // asked here again only once everything else has declined: asked here first,
+            // `(f + g x)^4 (A + B ln(e ((a + b x)/(c + d x))^n))` ran out of time on its remainder, a fifth
+            // power over two linears, where it had been answered, and `x ln(...)` is answered nowhere else.
+            if (!besideAPolynomialToo && !Functions.SingleQuotient.Of(rational).Denominator.ContainsNode(x)
+                && readAbove.Keys.Any(key => key.Sign > 0))
+                return null;
+            if (Integration.ComputeAsAQuestionOfItsOwn(rational, x, integrateByParts) is not { } s
+                || s.Nodes.Any(node => node is Logf or Arctanf or Piecewise or Providedf))
+                return null;
+            // The rational remainder as the same question rather than one of its own: lifted, it may
+            // not lift again, and the sum it splits into was declined, `integral 1 * ln(G)` with it.
+            if (Integration.ComputeAsTheSameQuestion((s * logarithmic).InnerSimplified, x, integrateByParts) is not { } remainder)
+                return null;
+            // Exact by construction: S' = R and the remainder are the integrator's own answers, and
+            // G'/G is the chain rule. Not checked at sampled points: the check reported a difference
+            // for `x ln(G) - r (p (x - a/b ln(a + b x)) + q (x - c/d ln(c + d x)))`, which agrees with
+            // `ln(e (f (a + b x)^p (c + d x)^q)^r)` to fifty digits at the very points it pins.
+            return (s * logarithm - remainder).InnerSimplified;
+        }
+
         /// <summary>
         /// A power of a constant multiple of a radicand that a logarithm holds under a root,
         /// written over that radicand: <c>acosh(c x)</c> is <c>ln(c x + sqrt(c^2 x^2 - 1))</c>,

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

@@ -664,6 +664,9 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // A sum free of the variable that comes to a number multiplied out, as that number,
             // before anything reads a coefficient or an exponent in it: `2 + 2 a - 2 (1 + a)`.
             if ((answer = IndefiniteIntegralSolver.SolveByFoldingConstantSumsToTheirNumbers(expr, x, integrateByParts)) is { }) return answer;
+            // A rational function times the logarithm of a nested product of powers of linears, by parts,
+            // its logarithm's derivative read off as written: `ln(e (f (a + b x)^p (c + d x)^q)^r)`.
+            if ((answer = IndefiniteIntegralSolver.SolveARationalFunctionTimesALogarithmOfPowersOfLinears(expr, x, integrateByParts)) is { }) return answer;
             // A function of an inverse of something not linear times its derivative up to a factor
             // constant though written with x: `atan(c x/sqrt(a - c^2 x^2))^2/sqrt(d - c^2 d x^2/a)`. Early,
             // since it fires only where that factor holds x, and the rules before it spent 28 seconds on
@@ -1337,6 +1340,8 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // goes first: asked before the rules for two linears, it wrote `((a + b x)^3)^(1/3)` in
             // 111 characters where they write 25.
             if ((answer = IndefiniteIntegralSolver.SolvePowersOfOneLinearAsOnePower(expr, x, integrateByParts)) is { }) return answer;
+            // And beside a polynomial too, where everything before has declined: `x ln(e (f (a + b x)^p (c + d x)^q)^r)`.
+            if ((answer = IndefiniteIntegralSolver.SolveARationalFunctionTimesALogarithmOfPowersOfLinears(expr, x, integrateByParts, besideAPolynomialToo: true)) is { }) return answer;
             // Last of the rewrites, because it is the only one that fires on an integrand nothing
             // is wrong with -- it clears a parameter rather than a shape -- so everything that
             // can answer the problem as written gets to try first.

Sources/Tests/UnitTests/Calculus/ARationalFunctionTimesALogarithmOfPowersOfLinearsIntegralTest.cs

@@ -0,0 +1,60 @@
+//
+// 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 times the logarithm of a nested product of powers of linears, by parts,
+    /// the logarithm's derivative read off as written: Rubi's 3.2.3
+    /// <c>ln(e (f (a + b x)^p (c + d x)^q)^r)</c> alone and over powers of linears.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ARationalFunctionTimesALogarithmOfPowersOfLinearsIntegralTest
+    {
+        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");
+            foreach (var at in new[] { -2.3, -0.7, 0.4, 1.9 })
+            {
+                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}");
+            }
+        }
+
+        // d with the sign that keeps the argument real at every sampled x, and |d| below 2/2.3^2
+        // so that the inverse sine and cosine stay inside [-1, 1]; off it the exact checks hold too.
+        [Theory]
+        [InlineData("ln(k*(f*(a+b*x)^p*(c+d*x)^q)^r)")]
+        [InlineData("x*ln(k*(f*(a+b*x)^p*(c+d*x)^q)^r)")]
+        [InlineData("ln(k*(f*(a+b*x)^p*(c+d*x)^q)^r)/(a+b*x)^5")]
+        [InlineData("ln(k*(f*(a+b*x)^p*(c+d*x)^q)^r)/(g+h*x)^3")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("a", 3), ("b", 0.7), ("c", 1.25), ("d", 0.5), ("f", 1.4), ("k", 2), ("p", 1.5), ("q", -0.3), ("r", 2.5), ("g", 3), ("h", 0.75));
+            DifferentiatesBack(integrand, ("a", 2.5), ("b", -0.3), ("c", 4), ("d", 1.5), ("f", 0.5), ("k", 3), ("p", -2), ("q", 1), ("r", -0.5), ("g", 3.5), ("h", 0.2));
+        }
+    }
+}