A power of the cosine beside a secant binomial is integrated in the cosine

BREAKING-CHANGES.md

@@ -347,6 +347,25 @@ 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 power of the cosine beside a secant binomial is integrated in the cosine
+
+**Answers where there were none.** `cos(c + d x)^2/(a + b sec(c + d x))` was declined, and so were the cosine's powers
+beside `A + B sec + C sec^2` over the same binomial (Rubi's 4.5.1.2, 4.5.3.1 and 4.5.4.2). Wherever the secant is
+defined, the integrand is one quotient in the cosine, `cos(c + d x)^3/(a cos(c + d x) + b)`, and it is integrated
+in that spelling now. The same holds for a power of the sine beside a cosecant binomial. Only for the binomial to
+the first power: for a higher power the secant's own rules answer
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"cos(c + d*x)^2/(a + b*sec(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | 1,091 characters |
+| `"cos(c + d*x)^2*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | 4,347 characters |
+| `"cos(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | 3,999 characters |
+| `"cos(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | 4,349 characters |
+| `"cos(c + d*x)^4*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | 5,579 characters |
+| `"cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | 12,065 characters |
+| `"sin(x)^2/(a + b*csc(x))".ToEntity().Integrate("x")` | `integral(...)` | 898 characters |
+
 ### One linear twice is not two linear powers
 
 **Answers that had no value.** `(a c + b c x)^(-3 - 2 p) (f + g x) (a^2 + 2 a b x + b^2 x^2)^p` was answered

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

@@ -28951,12 +28951,14 @@ bool IsAPolynomialInBoth(Entity polynomial)
         /// <remarks>
         /// The half-angle substitution read the secant's spelling into a rational function of the
         /// half-angle tangent that a substitution search expanded past thirty seconds, where the
-        /// cosine's is answered in half of one. Only for two sums or more, all below the bar. Rubi's 4.5.2.3.
+        /// cosine's is answered in half of one. Only for two sums or more, all below the bar, or beside the
+        /// cosine itself. Rubi's 4.5.2.3 and 4.5.4.2.
         /// https://github.com/asc-community/AngouriMath/issues/718
         /// </remarks>
         internal static Entity? SolveByWritingARationalFunctionOfTheSecantInTheCosine(Entity expr, Entity.Variable x, bool integrateByParts)
         {
             Entity? reciprocal = null;
+            var partners = new List<Entity>();
             foreach (var node in expr.Nodes)
             {
                 if (!node.ContainsNode(x) || node is Entity.Variable)
@@ -28969,6 +28971,9 @@ bool IsAPolynomialInBoth(Entity polynomial)
                         else if (reciprocal != node)
                             return null;
                         break;
+                    case Cosf or Sinf:
+                        partners.Add(node);
+                        break;
                     case Sumf or Minusf or Mulf or Divf or Powf(_, Number.Integer):
                         break;
                     default:
@@ -28982,15 +28987,29 @@ bool IsAPolynomialInBoth(Entity polynomial)
             }
             if (reciprocal is null || !TreeAnalyzer.TryGetPolyLinear(reciprocal.DirectChildren.First(), x, out _, out _))
                 return null;
+            var argument = reciprocal.DirectChildren.First();
+            var function = reciprocal is Secantf ? MathS.Cos(argument) : MathS.Sin(argument);
+            // The cosine of the same argument beside the secant, or the sine beside the cosecant, is
+            // what the rewritten integrand is in already; any other is a second function.
+            if (partners.Any(partner => partner != function))
+                return null;
             // Only two sums in the secant or more, all below the bar: one, or one above, the rules for
             // the secant answer as written, and in the cosine `sec(x)^5/(a + b sec(x))^3` ran past five
-            // seconds where it is half of one.
+            // seconds where it is half of one. Beside the cosine itself, though, the secant's rules have
+            // two functions to read, and in the cosine alone it is one quotient: the half-angle
+            // substitution wrote `cos(x)^2/(a + b sec(x))` in 117,409 characters where
+            // `cos(x)^3/(a cos(x) + b)` is 873. Rubi's 4.5.4.2.
             var (above, below) = Functions.SingleQuotient.Of(expr);
             bool ASum(Entity node) => node is Sumf or Minusf && node.ContainsNode(reciprocal);
-            if (above.Nodes.Any(ASum) || below.Nodes.Where(ASum).Distinct().Count() < 2)
+            if (partners.Count == 0 && (above.Nodes.Any(ASum) || below.Nodes.Where(ASum).Distinct().Count() < 2))
+                return null;
+            // Beside the cosine, a sum in the secant below, each to the first power only:
+            // `cos(x)/(a + b sec(x))^4` in the cosine is a partial fraction search past five seconds,
+            // which the secant's rules answer in a third of one, and with the sum in the sine instead,
+            // `csc(x)^3/(a + b sin(x)^3)` is a question for the rules of the sine's binomial.
+            if (partners.Count > 0
+                && (!below.Nodes.Any(ASum) || Mulf.LinearChildren(below).Any(factor => factor is Powf(var @base, _) && ASum(@base))))
                 return null;
-            var argument = reciprocal.DirectChildren.First();
-            var function = reciprocal is Secantf ? MathS.Cos(argument) : MathS.Sin(argument);
             var rewritten = Functions.SingleQuotient.Combine(expr.Replace(node => node == reciprocal ? 1 / function : node));
             return Integration.ComputeAsTheSameQuestion(rewritten, x, integrateByParts);
         }

Sources/Tests/UnitTests/Calculus/ACosineBesideASecantBinomialIntegralTest.cs

@@ -0,0 +1,47 @@
+//
+// 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 the cosine beside a rational function of the secant of the same argument, written
+    /// in the cosine alone, where it is one quotient. Rubi's 4.5.4.2.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ACosineBesideASecantBinomialIntegralTest
+    {
+        [Theory]
+        [InlineData("cos(c + d*x)^2*(A + K*sec(c + d*x)^2)/(a + b*sec(c + d*x))", 6000)]
+        [InlineData("cos(c + d*x)^3*(B*sec(c + d*x) + K*sec(c + d*x)^2)/(a + b*sec(c + d*x))", 6000)]
+        [InlineData("cos(c + d*x)^2*(A + B*sec(c + d*x) + K*sec(c + d*x)^2)/(a + b*sec(c + d*x))", 15000)]
+        [InlineData("cos(x)^2/(a + b*sec(x))", 1500)]
+        [InlineData("sin(x)^2/(a + b*csc(x))", 1500)]
+        public void DifferentiatesBack(string integrand, int atMostCharacters)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Assert.True(integral.Stringize().Length <= atMostCharacters, $"{integral.Stringize().Length} characters");
+            Entity Pinned(Entity e) => e.Substitute("a", 2).Substitute("b", 3).Substitute("c", "1/4".ToEntity()).Substitute("d", "3/2".ToEntity())
+                .Substitute("A", 5).Substitute("B", 7).Substitute("K", 11);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            foreach (var at in new[] { "0.35", "0.6", "1.1", "1.4" })
+            {
+                var want = original.Substitute("x", at.ToEntity()).EvalNumerical();
+                var got = derivative.Substitute("x", at.ToEntity()).EvalNumerical();
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart)
+                            < 1e-9 * Math.Max(1, Math.Abs((double)want.RealPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+        }
+    }
+}