Ask a hyperbolic quotient over a quadratic in its sine or cosine as the sine's, before the tangent

BREAKING-CHANGES.md

@@ -1004,6 +1004,21 @@ roots, `1 - x^2`, is taken as its two ([#718](https://github.com/asc-community/A
 | `"1/((1 - x^2)*(a + b*x^3))".ToEntity().Integrate("x")` | `integral(...)` | 1,600 characters |
 | `"1/(x*(1 - x^2)*(a + b*x^3))".ToEntity().Integrate("x")` | `integral(...)` | 1,637 characters |
 
+### A hyperbolic quotient over a quadratic in its sine or cosine is asked as the sine's, before the tangent
+
+**Answers where there were none, and one longer spelling.** Rubi's 6.7.1 `sinh(x)^2/(a + b sinh(x) + c sinh(x)^2)`
+and `sinh(x)^3/(a + b sinh(x) + c sinh(x)^2)` ran past the budget. The tangent of the half argument made each a
+quartic with symbols in it, and its partial fractions did not return. Under Osborn's rule each is a rational
+function of a sine over a symbolic quadratic in it, and the sine's rule for that shape is now asked first, before
+the tangent. A quotient master answered some other way can come back spelled through the sine instead
+([#233](https://github.com/asc-community/AngouriMath/issues/233)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sinh(x)^2/(a+b*sinh(x)+c*sinh(x)^2)".ToEntity().Integrate("x")` | no answer in twenty seconds | 681 characters |
+| `"sinh(x)^3/(a+b*sinh(x)+c*sinh(x)^2)".ToEntity().Integrate("x")` | no answer in twenty seconds | 810 characters |
+| `"cosh(x)^2/(a+b*cosh(x)+c*cosh(x)^2)".ToEntity().Integrate("x")` | no answer in twenty seconds; 791 characters on the unreleased master | 813 characters |
+
 ### A hyperbolic integrand is asked as a trigonometric one
 
 **Answers where there were none.** The parser writes the hyperbolic functions as exponentials, so the trigonometric

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

@@ -1429,7 +1429,8 @@ static bool APowerOfAQuotientOfLinearsVanishingAtPlusOrMinusOne(Entity inU, Vari
         /// the bar of degree above two: the second pass answers it in <c>e^L</c>, and the question in
         /// <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)
+        internal static Entity? SolveAHyperbolicIntegrandAsATrigonometricOne(Entity expr, Entity.Variable x, bool integrateByParts, bool rationalToo,
+            bool overAQuadraticInTheSine = 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.
@@ -1497,7 +1498,11 @@ static bool APowerOfAQuotientOfLinearsVanishingAtPlusOrMinusOne(Entity inU, Vari
                 && AFactorAboveAQuadraticInS(inS, s))
                 return null;
             var question = MathS.i / slopeOfL * inS;
-            if (Integration.ComputeAsAQuestionOfItsOwn(question, s, integrateByParts) is not { } inTermsOfS
+            // 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
+                    ? SolveARationalFunctionOfASineOverASymbolicQuadratic(inS, s, integrateByParts) is { } overTheQuadratic ? MathS.i / slopeOfL * overTheQuadratic : null
+                    : Integration.ComputeAsAQuestionOfItsOwn(question, s, integrateByParts)) is not { } inTermsOfS
                 || inTermsOfS.Nodes.Any(node => node is Integralf || node == MathS.NaN))
                 return null;
             var linear = interceptOfL.Evaled is Number.Complex { IsZero: true } ? slopeOfL * x : interceptOfL + slopeOfL * x;

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

@@ -1229,6 +1229,11 @@ internal static bool ASumOverRootsWouldAnswer
             // `cosh(x)^7/(a + b cosh(x)^2)` and `sech(x)^3/(a + b sinh(x))^2` are a second here
             // and past the budget there.
             if ((answer = IndefiniteIntegralSolver.SolveByBiochesOddHyperbolicSubstitution(expr, x, integrateByParts)) is { }) return answer;
+            // A rational function of e^L that Osborn's rule makes one of a sine or a cosine over a symbolic
+            // quadratic in it, asked as that before the tangent: under it Rubi's
+            // `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;
             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/AHyperbolicQuotientOverAQuadraticInItAsTheSinesIntegralTest.cs

@@ -0,0 +1,53 @@
+//
+// 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 System.Threading;
+using System.Threading.Tasks;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// A rational function of a hyperbolic sine or cosine over a symbolic quadratic in it, asked under
+    /// Osborn's rule as one of a sine over a quadratic before the tangent of the half argument: under that
+    /// tangent Rubi's 6.7.1 <c>sinh(x)^2/(a + b sinh(x) + c sinh(x)^2)</c> is a quartic with symbols in it,
+    /// and its partial fractions ran past the budget.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AHyperbolicQuotientOverAQuadraticInItAsTheSinesIntegralTest
+    {
+        [Theory]
+        [InlineData("sinh(x)^2/(a + b*sinh(x) + c*sinh(x)^2)")]
+        [InlineData("sinh(x)^3/(a + b*sinh(x) + c*sinh(x)^2)")]
+        [InlineData("cosh(x)^2/(a + b*cosh(x) + c*cosh(x)^2)")]
+        public void IsAnswered(string integrand)
+        {
+            Entity? answer = null;
+            // A thread of its own, so that the guard is not spent waiting behind other tests' work.
+            var task = Task.Factory.StartNew(() => answer = integrand.ToEntity().Integrate("x"),
+                CancellationToken.None, TaskCreationOptions.LongRunning, TaskScheduler.Default);
+            Assert.True(task.Wait(LimitTermination.Guard), $"{integrand} did not finish within {LimitTermination.Guard}");
+            var integral = answer!;
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.4);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            foreach (var at in new[] { -1.7, -0.6, 0.4, 1.3, 2.2 })
+            {
+                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}");
+            }
+        }
+    }
+}