Ask a hyperbolic quotient over a linear in its cosine as the cosine's, before the tangent

BREAKING-CHANGES.md

@@ -561,6 +561,22 @@ power, which by parts answers as before ([#233](https://github.com/asc-community
 | `"x*(a+b*atanh(c*x))^2/(d+c*d*x)^3".ToEntity().Integrate("x")` | `integral(...)` | 731 characters |
 | `"x*(a+b*acoth(c*x))^2/(d+c*d*x)^3".ToEntity().Integrate("x")` | `integral(...)` | 756 characters |
 
+### A hyperbolic quotient over a linear in its cosine is asked as the cosine's
+
+**Answers where there were none, and shorter than master's.** Rubi's 6.2.5 `sech(x)^3/(a + b cosh(x))` and 6.5.3
+`sech(x)^4/(a + b sech(x))` ran past the corpus's budget. Under the tangent of the half argument each is a cube of
+`1 + t^2` beside a symbolic quadratic, and was answered in about 118,000 characters. By Osborn's rule each is a rational
+function of a cosine alone, over a sum linear in it, and that question is now asked first. In the cosine,
+`1/(cos(s)^3 (a + b cos(s)))` is a partial fraction in it
+([#233](https://github.com/asc-community/AngouriMath/issues/233)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sech(x)^3/(a+b*cosh(x))".ToEntity().Integrate("x")` | `integral(...)` | 2,396 characters, where before this change master gave 118,366 |
+| `"sech(x)^4/(a+b*sech(x))".ToEntity().Integrate("x")` | `integral(...)` | 2,412 characters, where before this change master gave 118,032 |
+| `"cosh(x)^3/(a+b*cosh(x))".ToEntity().Integrate("x")` | `integral(...)` | 1,108 characters, where before this change master gave 2,858 |
+| `"cosh(x)^2/(a+b*sech(x))".ToEntity().Integrate("x")` | `integral(...)` | 802 characters, where before this change master gave 2,820 |
+
 ### An arctangent over a power of `d (1 + i c x)` is integrated under the tangent
 
 **Answers where there were none.** `(a + b atan(c x))^2/(d + i c d x)^2` ran out of time, and so did seven more of

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

@@ -1430,7 +1430,7 @@ static bool APowerOfAQuotientOfLinearsVanishingAtPlusOrMinusOne(Entity inU, Vari
         /// <c>s</c> took seconds to decline, five gigabytes for <c>1/(1 + sinh(x)^3)</c> against 0.2.
         /// </remarks>
         internal static Entity? SolveAHyperbolicIntegrandAsATrigonometricOne(Entity expr, Entity.Variable x, bool integrateByParts, bool rationalToo,
-            bool overAQuadraticInTheSine = false)
+            bool overAQuadraticInTheSine = false, bool overALinearInTheCosine = false)
         {
             // Only the question asked: a question in s is not cheap to decline, and below the top every
             // sub-question a route gives up on would ask it.
@@ -1498,6 +1498,12 @@ static bool APowerOfAQuotientOfLinearsVanishingAtPlusOrMinusOne(Entity inU, Vari
                 && AFactorAboveAQuadraticInS(inS, s))
                 return null;
             var question = MathS.i / slopeOfL * inS;
+            // Or, asked as early, only a rational function of the cosine alone over a sum linear in it, or in
+            // the secant: under the tangent of the half argument Rubi's `sech(x)^3/(a + b cosh(x))` is a
+            // cube of `1 + t^2` beside a symbolic quadratic, answered in 118,366 characters, where in the
+            // cosine `1/(cos(s)^3 (a + b cos(s)))` is a partial fraction in it.
+            if (overALinearInTheCosine && !IsARationalFunctionOfTheCosineOverALinearInIt(inS, s))
+                return null;
             // Or the one rule for a rational function of a sine or a cosine over a symbolic quadratic in
             // it, asked ahead of the exponential substitution and cheap to decline.
             if ((overAQuadraticInTheSine
@@ -1520,6 +1526,26 @@ static bool APowerOfAQuotientOfLinearsVanishingAtPlusOrMinusOne(Entity inU, Vari
             }).Substitute(s, -MathS.i * linear).InnerSimplified;
             return Functions.PartialFractions.DerivativeHoldsAtSampledPoints(answer, expr, x) ? answer : null;
 
+            // Whether inS holds s only in cos(s), and below the bar a sum linear in it, to the first power:
+            // over `b^2 + 2 a b cos(s) + a^2 cos(s)^2`, the square of one, the question in s took four times
+            // the memory the tangent did, Rubi's 6.7.1 `(a + b cosh(x))/(b + a cosh(x))^2`.
+            static bool IsARationalFunctionOfTheCosineOverALinearInIt(Entity inS, Variable s)
+            {
+                var cosine = MathS.Cos(s);
+                foreach (var node in inS.Nodes)
+                    if (node.ContainsNode(s) && node != cosine && node != s
+                        && node is not (Sumf or Minusf or Mulf or Divf or Powf(_, Number.Integer)))
+                        return false;
+                var (_, below) = Functions.SingleQuotient.Of(inS);
+                var y = Variable.CreateUnique(inS, "y_osb");
+                bool ASum(Entity node) => node is Sumf or Minusf && node.ContainsNode(cosine);
+                bool Linear(Entity sum) => TreeAnalyzer.TryGetPolynomial(sum.Replace(node => node == cosine ? y : node), y, out var read)
+                    && read.Keys.Max()!.Equals(EInteger.One);
+                var sums = Mulf.LinearChildren(below).Where(ASum).ToList();
+                return sums.Count > 0 && sums.All(Linear)
+                    && !Mulf.LinearChildren(below).Any(factor => factor is Powf(var @base, _) && ASum(@base));
+            }
+
             // Whether inS, rational in cos(s) alone or sin(s) alone, has a denominator of degree three or
             // more that t = tan(s/2) or t = tan(s) leaves with an irreducible factor of degree above two,
             // which the rational rules write only as a sum over roots. The cosine, and the square of the

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

@@ -1242,6 +1242,9 @@ internal static bool ASumOverRootsWouldAnswer
             // `sinh(x)^2/(a + b sinh(x) + c sinh(x)^2)` is a quartic with symbols, and its partial
             // fractions ran past the budget, where the sine's rule answers the question in s at once.
             if ((answer = IndefiniteIntegralSolver.SolveAHyperbolicIntegrandAsATrigonometricOne(expr, x, integrateByParts, rationalToo: true, overAQuadraticInTheSine: true)) is { }) return answer;
+            // And one of the cosine alone over a sum linear in it, which the tangent made a cube of
+            // `1 + t^2` beside a symbolic quadratic: `sech(x)^3/(a + b cosh(x))` in 118,366 characters.
+            if ((answer = IndefiniteIntegralSolver.SolveAHyperbolicIntegrandAsATrigonometricOne(expr, x, integrateByParts, rationalToo: true, overALinearInTheCosine: true)) is { }) return answer;
             if ((answer = IndefiniteIntegralSolver.SolveARationalFunctionOfTheHyperbolicTangent(expr, x, integrateByParts)) is { }) return answer;
             // A rational function of a sine or a cosine over a quadratic in it with a symbol
             // among the coefficients, split over the quadratic's two roots: under the

Sources/Tests/UnitTests/Calculus/AHyperbolicQuotientOverALinearInItsCosineIntegralTest.cs

@@ -0,0 +1,51 @@
+//
+// 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 of the hyperbolic cosine alone over a sum linear in it, or in the secant,
+    /// asked by Osborn's rule as one of the cosine before the tangent of the half argument: under that
+    /// tangent Rubi's 6.2.5 <c>sech(x)^3/(a + b cosh(x))</c> and 6.5.3 <c>sech(x)^4/(a + b sech(x))</c>
+    /// are a cube of <c>1 + t^2</c> beside a symbolic quadratic, and were answered in 118,366 and
+    /// 118,032 characters, past the corpus's budget.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AHyperbolicQuotientOverALinearInItsCosineIntegralTest
+    {
+        [Theory]
+        // Rubi 6.2.5:89.
+        [InlineData("sech(x)^3/(a + b*cosh(x))")]
+        // Rubi 6.5.3:139.
+        [InlineData("sech(x)^4/(a + b*sech(x))")]
+        [InlineData("cosh(x)^2/(a + b*sech(x))")]
+        public void IsAnsweredInTheCosine(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 5000, $"{integrand} came back in {text.Length} characters");
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7);
+            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}");
+            }
+        }
+    }
+}