An even rational function with symbols skips the substitution search

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

@@ -21546,6 +21546,26 @@ private static bool IsProperByDegree(Entity numerator, Entity denominator, Entit
                && TreeAnalyzer.TryGetPolynomial(denominator, x, out var below) && below.Count > 0
                && above.Keys.Max()!.CompareTo(below.Keys.Max()!) < 0;
 
+        /// <summary>Whether <paramref name="x"/> stands in <paramref name="expr"/> only under even whole powers.</summary>
+        private static bool IsEvenAsWritten(Entity expr, Entity.Variable x)
+            => expr.Nodes.Count(node => node == x)
+               == expr.Nodes.Count(node => node is Powf(var @base, Number.Integer power) && @base == x && power.EInteger.IsEven);
+
+        /// <summary>
+        /// The degree in <paramref name="x"/> of a polynomial as it is written, without expanding
+        /// it: of a power its base's times the exponent, of a product the sum, of a sum the most.
+        /// </summary>
+        private static int DegreeAsWritten(Entity expr, Entity.Variable x) => expr switch
+        {
+            _ when !expr.ContainsNode(x) => 0,
+            Variable => 1,
+            Powf(var @base, Number.Integer power) when power.EInteger.CanFitInInt32() => DegreeAsWritten(@base, x) * power.EInteger.ToInt32Checked(),
+            Mulf(var left, var right) => DegreeAsWritten(left, x) + DegreeAsWritten(right, x),
+            Sumf(var left, var right) => System.Math.Max(DegreeAsWritten(left, x), DegreeAsWritten(right, x)),
+            Minusf(var left, var right) => System.Math.Max(DegreeAsWritten(left, x), DegreeAsWritten(right, x)),
+            _ => 0,
+        };
+
         /// <summary>Whether every node of <paramref name="expr"/> holding <paramref name="x"/> is a sum, a product, a quotient or a whole power.</summary>
         private static bool IsARationalFunction(Entity expr, Entity.Variable x)
             => expr.Nodes.All(node => !node.ContainsNode(x) || node is Variable or Sumf or Minusf or Mulf or Divf || node is Powf(_, Number.Integer));
@@ -28928,6 +28948,15 @@ private static bool IsAFunctionOfTheTangentAlone(Entity expr, Entity.Variable u)
             // page of piecewise on the discriminant of the quadratic the other two make.
             if (IsARationalFunction(expr, x) && TryReadAsQuotient(expr, out _, out var writtenBelow) && IsAProductOfSymbolicLinearFactors(writtenBelow, x))
                 return null;
+            // And an even one with symbols in it, past a sextic below the bar: the derivative of
+            // anything even is odd, so no polynomial candidate is the substitution for it, and
+            // the search spent thirty seconds simplifying the quotient by each to decline them all
+            // on `(1 + x^2)^2/((c + d + (c + 3d) x^2 + 2d x^4)^3 (1 + 2x^2))`, which the half-angle
+            // tangent makes of `1/((c + d sec(y))^3 sqrt(a + a sec(y)))`, where the partial
+            // fractions answer it in under one.
+            if (IsARationalFunction(expr, x) && IsEvenAsWritten(expr, x) && expr.Nodes.Any(node => node is Variable symbol && symbol != x)
+                && TryReadAsQuotient(expr, out _, out var evenBelow) && DegreeAsWritten(evenBelow, x) >= 6)
+                return null;
             // A rational function of exponentials of linears in x with a whole power of a
             // sum of them in it is the exponential substitution's, exactly and at once, and
             // this search is not the tool for it: `tanh(x)^5/sech(x)^4` arrives as a fifth

Sources/Tests/UnitTests/Calculus/AnEvenRationalFunctionIntegralTest.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>
+    /// An even rational function with symbols in it and a denominator past a sextic, by the
+    /// partial fractions without the substitution search in front of them. Rubi's 1.2.2.6, and
+    /// what the half-angle tangent makes of 4.5.2.1's.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnEvenRationalFunctionIntegralTest
+    {
+        [Theory]
+        [InlineData("(1 + x^2)^2/((c + d + (c + 3*d)*x^2 + 2*d*x^4)^3*(1 + 2*x^2))")]
+        [InlineData("(d + g*x^2 + f*x^4)/(x^4*(a + b*x^2 + c*x^4)^2)")]
+        public void ByThePartialFractions(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 40000, $"{text.Length} characters of answer for {integrand}");
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.4).Substitute("c", 0.9).Substitute("d", 0.6).Substitute("f", 0.7).Substitute("g", 0.2);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -2.1, -0.4, 0.3, 0.6, 1.6, 2.5 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                if (want.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}");
+        }
+    }
+}