Write an even power of the sine beside the secant in the cosine

BREAKING-CHANGES.md

@@ -117,6 +117,21 @@ coefficients are the roots of that quadratic, and the polynomial over it is spli
 | `"1/(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)".ToEntity().Integrate("x")` | `integral(...)` | 3,920 characters |
 | `"x/(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)".ToEntity().Integrate("x")` | `integral(...)` | 2,315 characters |
 
+### An even power of the sine beside the secant is written in the cosine
+
+**Answers where there were none, shorter than master's.** Rubi's 4.5.1.3 `sin(c + d x)^2/(a + b sec(c + d x))` ran
+past the corpus's budget. The half-angle substitution wrote it in 117,751 characters, and `sin(x)^4/(a + b sec(x))` in
+160,657. A rational function of the secant was written in the cosine only beside the cosine itself. The sine beside it
+stands to even powers only, and `sin(z)^(2k)` is `(1 - cos(z)^2)^k`, so the integrand is now written in the cosine, as a
+polynomial over a linear in it. The cosine beside the cosecant is written in the sine the same way
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sin(c+d*x)^2/(a+b*sec(c+d*x))".ToEntity().Integrate("x")` | `integral(...)` | 1,187 characters, where before this change master gave 117,751 |
+| `"sin(x)^4/(a+b*sec(x))".ToEntity().Integrate("x")` | `integral(...)` | 1,729 characters, where before this change master gave 160,657 |
+| `"cos(x)^2/(a+b*csc(x))".ToEntity().Integrate("x")` | `integral(...)` | 994 characters, where before this change master gave 2,271 |
+
 ### An exponential of a quadratic over a power of the linear through its vertex is a moment about it
 
 **Answers where there were none.** `f^(a + b x + c x^2)/(b + 2 c x)` was declined, and so were five more of Rubi's 2.3,

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

@@ -46958,8 +46958,14 @@ bool IsAPolynomialInBoth(Entity polynomial)
             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))
+            // what the rewritten integrand is in already, and the other of the two is too where it
+            // stands only to even powers, `sin(z)^(2k)` being `(1 - cos(z)^2)^k`: the half-angle
+            // substitution wrote `sin(x)^2/(a + b sec(x))` in 117,517 characters, where
+            // `(1 - cos(x)^2) cos(x)/(a cos(x) + b)` is 969. Any other is a second function.
+            var complement = reciprocal is Secantf ? MathS.Sin(argument) : MathS.Cos(argument);
+            static bool AnEvenPower(Entity node, Entity of) => node is Powf(var @base, Number.Integer { EInteger: var power }) && @base == of && power.IsEven && power.Sign > 0;
+            var inEvenPowers = expr.Nodes.Count(node => node == complement) == expr.Nodes.Count(node => AnEvenPower(node, complement));
+            if (partners.Any(partner => partner != function && (partner != complement || !inEvenPowers)))
                 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
@@ -46978,7 +46984,11 @@ bool IsAPolynomialInBoth(Entity polynomial)
             if (partners.Count > 0
                 && (!below.Nodes.Any(ASum) || Mulf.LinearChildren(below).Any(factor => factor is Powf(var @base, _) && ASum(@base))))
                 return null;
-            var rewritten = Functions.SingleQuotient.Combine(expr.Replace(node => node == reciprocal ? 1 / function : node));
+            var rewritten = Functions.SingleQuotient.Combine(expr.Replace(node =>
+                node == reciprocal ? 1 / function
+                : AnEvenPower(node, complement) && node is Powf(_, Number.Integer { EInteger: var power })
+                    ? MathS.Pow(1 - MathS.Pow(function, 2), Number.Integer.Create(power / 2))
+                    : node));
             return Integration.ComputeAsTheSameQuestion(rewritten, x, integrateByParts);
         }
 

Sources/Tests/UnitTests/Calculus/AnEvenPowerOfTheSineBesideASecantIntegralTest.cs

@@ -0,0 +1,49 @@
+//
+// 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 power of the sine beside a rational function of the secant, written in the cosine,
+    /// <c>sin(z)^(2k)</c> as <c>(1 - cos(z)^2)^k</c>: the half-angle substitution wrote Rubi's 4.5.1.3
+    /// <c>sin(c + d x)^2/(a + b sec(c + d x))</c> in 117,751 characters and past the corpus's budget,
+    /// where in the cosine it is a polynomial over a linear. The cosine beside the cosecant the same way.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnEvenPowerOfTheSineBesideASecantIntegralTest
+    {
+        [Theory]
+        // Rubi 4.5.1.3:238.
+        [InlineData("sin(c + d*x)^2/(a + b*sec(c + d*x))")]
+        [InlineData("sin(x)^4/(a + b*sec(x))")]
+        [InlineData("cos(x)^2/(a + b*csc(x))")]
+        public void IsAnsweredInTheCosine(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 3000, $"{integrand} came back in {text.Length} characters");
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.4).Substitute("d", 0.9);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            foreach (var at in new[] { -2.2, -0.9, 0.3, 1.4, 2.1 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                Assert.False(got.IsNaN, $"d/dx of the antiderivative of {integrand} has no value at x = {at}, where the integrand is {want}");
+                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}");
+            }
+        }
+    }
+}