Integrate a rational function over an inverse tangent's quadratic beside it by partial fractions

BREAKING-CHANGES.md

@@ -310,6 +310,20 @@ carried the discriminant's being real in its condition, and held nowhere when it
 | `"piecewise(2 * x provided not x = 0, 3 * x ^ 2 provided i - 1 < 0)".ToEntity().DomainCondition` | `false` | `not x = 0` |
 | `"1/(sqrt(a + i*a*x)*(c + d*x)^(3/2)*(a - i*a*x))".ToEntity().Integrate("x")` | `integral(...)`; on the unreleased master, an answer with no value at any point | 3,267 characters, defined |
 
+### A rational function over the quadratic of an inverse tangent beside it is split
+
+**Answers where there were none.** `x^4 atan(a x)/(c + a^2 c x^2)` ran out of time, and so did three more of Rubi's 5.3.4.
+The rational factor is now divided down and split over `c + a^2 c x^2` and the powers of x, and each part beside the
+inverse tangent is integrated on its own
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x^4*atan(a*x)/(c+a^2*c*x^2)".ToEntity().Integrate("x")` | `integral(...)` | 233 characters |
+| `"x^2*atan(a*x)/(c+a^2*c*x^2)".ToEntity().Integrate("x")` | `integral(...)` | 126 characters |
+| `"atan(a*x)/(x^2*(c+a^2*c*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 612 characters |
+| `"atan(a*x)/(x^4*(c+a^2*c*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 894 characters |
+
 ### A piecewise case whose condition is undefined is passed over
 
 **A different answer.** A case whose predicate evaluates to NaN -- an order comparison of a number

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

@@ -26445,6 +26445,101 @@ static void Add(Dictionary<int, ERational> into, int k, ERational value)
             return answer.Nodes.Any(node => node == MathS.NaN) ? null : answer;
         }
 
+        /// <summary>
+        /// A rational function over the quadratic an inverse tangent's derivative is written over,
+        /// times a function of that inverse tangent, <c>x^4 atan(a x)/(c + a^2 c x^2)</c>, with the
+        /// rational function split into partial fractions over its written factors and each piece
+        /// beside the function its own question: <c>atan(a x)/(1 + a^2 x^2)</c> is
+        /// <c>atan(a x)^2/(2a)</c>, and the rest is a power of x beside the inverse tangent.
+        /// </summary>
+        /// <remarks>
+        /// The rules before it timed out on Rubi's 5.3.4 <c>x^4 atan(a x)/(c + a^2 c x^2)</c> and
+        /// <c>atan(a x)/(x^4 (c + a^2 c x^2))</c>, and declined <c>x^2 atan(a x)/(c + a^2 c x^2)</c>.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </remarks>
+        internal static Entity? SolveARationalFunctionOverTheQuadraticOfAnArctangentBesideIt(Entity expr, Entity.Variable x)
+        {
+            if (!Integration.AnsweringTheQuestionAsked)
+                return null;
+            static IEnumerable<(Entity Factor, bool Below)> Factors(Entity e, bool below) => e switch
+            {
+                Mulf(var l, var r) => Factors(l, below).Concat(Factors(r, below)),
+                Divf(var l, var r) => Factors(l, below).Concat(Factors(r, !below)),
+                _ => new[] { (e, below) }
+            };
+            Entity? function = null;
+            Entity above = Number.Integer.One, below = Number.Integer.One;
+            foreach (var (factor, isBelow) in Factors(expr, false))
+            {
+                if (factor.Nodes.Any(node => node is Arctanf && node.ContainsNode(x)))
+                {
+                    if (function is not null || isBelow)
+                        return null;
+                    function = factor;
+                }
+                else if (isBelow)
+                    below *= factor;
+                else
+                    above *= factor;
+            }
+            if (function is null || !below.ContainsNode(x))
+                return null;
+            var arctangents = function.Nodes.Where(node => node is Arctanf && node.ContainsNode(x)).Distinct().ToList();
+            if (arctangents.Count != 1 || arctangents[0] is not Arctanf(var argument)
+                || !TreeAnalyzer.TryGetPolyLinear(argument, x, out var slope, out var intercept)
+                || !Functions.PartialFractions.IsZeroAsAValue(intercept))
+                return null;
+            // The rest is a rational function in x, and one written factor below the bar is a
+            // multiple of 1 + (a x)^2.
+            var quadratic = 1 + MathS.Sqr(slope) * MathS.Sqr(x);
+            var (numerator, denominator) = Functions.SingleQuotient.Of(above / below);
+            if (!IsARationalFunction(numerator / denominator, x)
+                || !Mulf.LinearChildren(denominator).Any(part => part.ContainsNode(x)
+                    && part.Substitute(x, Number.Integer.Zero).InnerSimplified is var atZero
+                    && !Functions.PartialFractions.IsZeroAsAValue(atZero)
+                    && Functions.PartialFractions.IsZeroAsAValue((part - atZero * quadratic).InnerSimplified)))
+                return null;
+            // An improper one divided first: `x^2/(c + a^2 c x^2)` is `1/(a^2 c) - 1/(a^2 (c + a^2 c x^2))`.
+            var pieces = new List<Entity>();
+            if (!IsProperByDegree(numerator, denominator, x))
+            {
+                if (TreeAnalyzer.PolynomialLongDivision(numerator, denominator, genericCase: true, inTermsOf: x) is not var (quotient, properPart))
+                    return null;
+                pieces.Add(quotient);
+                // The generic case, as the integrator answers it: the remainder carries the
+                // condition that the leading coefficient is not zero.
+                properPart = properPart.Replace(node => node is Providedf(var expression, _) ? expression : node).InnerSimplified;
+                (numerator, denominator) = Functions.SingleQuotient.Of(properPart);
+                if (Functions.PartialFractions.IsZeroAsAValue(numerator))
+                    numerator = Number.Integer.Zero;
+            }
+            Entity decomposition;
+            if (numerator == Number.Integer.Zero)
+                decomposition = Number.Integer.Zero;
+            else if (!Functions.PartialFractions.TrySplitOverWrittenFactors(numerator, denominator, x, out var split))
+            {
+                if (pieces.Count == 0)
+                    return null;
+                decomposition = numerator / denominator;
+            }
+            else
+                decomposition = split;
+            // The split comes back as one sum over a factor free of x, `(0 + 1/(c x^2) + ...)/1`.
+            var (terms, divisor) = decomposition is Divf(var sum, var free) && !free.ContainsNode(x) ? (sum, free) : (decomposition, (Entity)Number.Integer.One);
+            pieces.AddRange(Sumf.LinearChildren(terms).Where(term => !Functions.PartialFractions.IsZeroAsAValue(term)).Select(term => term / divisor));
+            if (pieces.Count < 2)
+                return null;
+            Entity answer = Number.Integer.Zero;
+            foreach (var piece in pieces)
+            {
+                if (Integration.ComputeAsAQuestionOfItsOwn((piece * function).InnerSimplified, x, integrateByParts: true) is not { } integrated)
+                    return null;
+                answer += integrated;
+            }
+            answer = answer.InnerSimplified;
+            return answer.Nodes.Any(node => node == MathS.NaN) ? null : answer;
+        }
+
         /// <summary>
         /// A rational function of a sine, or of a cosine, over a quadratic in it with a symbol
         /// among the coefficients: <c>sin(x)/(a + b sin(x) + c sin(x)^2)</c>, and the same with

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

@@ -680,6 +680,9 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // A power of a sine, cosine or hyperbolic one over a binomial in a power of it, over the
             // binomial's roots, before the rules that spent the budget on `sinh(y)^4/(a + b sinh(y)^3)`.
             if ((answer = IndefiniteIntegralSolver.SolveAPowerOverABinomialInAPowerOfASineOverItsRoots(expr, x)) is { }) return answer;
+            // A rational function over the quadratic of an inverse tangent beside it, split, before the
+            // rules that timed out on `x^4 atan(a x)/(c + a^2 c x^2)`.
+            if ((answer = IndefiniteIntegralSolver.SolveARationalFunctionOverTheQuadraticOfAnArctangentBesideIt(expr, x)) is { }) return answer;
             // A power of x times a power of a sum holding an inverse function of x^n, under t = x^n,
             // before the rules that timed out on `x^3 (a + b atanh(c x^2))^2`.
             if ((answer = IndefiniteIntegralSolver.SolveAPowerOfXTimesAFunctionOfAnotherUnderThatPower(expr, x)) is { }) return answer;

Sources/Tests/UnitTests/Calculus/ARationalFunctionOverTheQuadraticOfAnArctangentIntegralTest.cs

@@ -0,0 +1,59 @@
+//
+// 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 over the quadratic an inverse tangent's derivative is written over, beside
+    /// that inverse tangent, split into partial fractions: Rubi's 5.3.4 <c>x^4 atan(a x)/(c + a^2 c x^2)</c>
+    /// and <c>atan(a x)/(x^4 (c + a^2 c x^2))</c>, which timed out.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ARationalFunctionOverTheQuadraticOfAnArctangentIntegralTest
+    {
+        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}");
+            }
+        }
+
+        // The quadratic's factor is q, not Rubi's c: the helper takes C, and c is too close.
+        [Theory]
+        [InlineData("x^4*atan(a*x)/(q+a^2*q*x^2)")]
+        [InlineData("x^2*atan(a*x)/(q+a^2*q*x^2)")]
+        [InlineData("atan(a*x)/(x^2*(q+a^2*q*x^2))")]
+        [InlineData("atan(a*x)/(x^4*(q+a^2*q*x^2))")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("a", 1.3), ("q", 0.7));
+            DifferentiatesBack(integrand, ("a", -0.4), ("q", -2));
+        }
+    }
+}