Read csch as the difference of the exponentials with its half gathered elsewhere

BREAKING-CHANGES.md

@@ -521,6 +521,19 @@ 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 cosecant beside a power of a binomial in the sine is read through its exponentials
+
+**Answers where there were none.** `csch(c + d x) (a + b sinh(c + d x)^4)^2` ran past fifteen minutes. `csch(y)` arrives
+as `(e^y - e^-y)^(-1)` beside a 2 the product gathered, and the reader of the hyperbolic functions recognised the
+difference only with its half; it now reads `e^y - e^-y` as `2 sinh(y)` and `e^y + e^-y` as `2 cosh(y)` under a whole
+power, and the integrand is answered in a fraction of a second
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"csch(c+d*x)*(a+b*sinh(c+d*x)^4)^2".ToEntity().Integrate("x")` | past fifteen minutes | 426 characters |
+| `"csch(c+d*x)*(a+b*sinh(c+d*x)^4)^3".ToEntity().Integrate("x")` | past fifteen minutes | 646 characters |
+
 ### An exponential of a whole multiple of an inverse hyperbolic tangent is written algebraically first
 
 **Answers where there were none.** `e^(n artanh(u))` with u linear is written as powers of `1 + u` and `1 - u`, and then

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

@@ -29165,9 +29165,22 @@ private static (Entity OverTheTwo, Entity Argument)? ReadTheHyperbolicFunctions(
                             return -c / s;
                         case Divf(Number.Integer { EInteger: var m }, var inner) when m.Equals(EInteger.One) && ReadTheTwo(inner) is { } read:
                             return 1 / read;
+                        // The difference and the sum with their halves taken elsewhere, as a whole
+                        // power only: `csch(y)` arrives as `(e^y - e^-y)^(-1)` beside a 2 the
+                        // product gathered. Inside a fractional power, or below a constant other
+                        // than 1, the reading hands the caller a root it then declines.
+                        case Powf(var b, Number.Integer { EInteger: var n }) when ReadTheBare(b) is { } bare:
+                            return MathS.Pow(bare, n);
                     }
                     return null;
                 }
+                Entity? ReadTheBare(Entity node) => node switch
+                {
+                    Minusf(var a, var b) when Is(a, one) && Is(b, one?.Negate()) => 2 * s,
+                    Minusf(var a, var b) when Is(a, one?.Negate()) && Is(b, one) => -2 * s,
+                    Sumf(var a, var b) when Is(a, one) && Is(b, one?.Negate()) || Is(a, one?.Negate()) && Is(b, one) => 2 * c,
+                    _ => null
+                };
                 bool Is(Entity node, EInteger? k) => k is { } wanted && exponentials.TryGetValue(node, out var found) && found.Equals(wanted);
                 var read = expr.Replace(node => node.ContainsNode(x) && ReadTheTwo(node) is { } inTheTwo ? inTheTwo : node);
                 if (read.ContainsNode(s) || read.ContainsNode(c))

Sources/Tests/UnitTests/Calculus/AnOddHyperbolicIntegrandWithAPowerOfABinomialIntegralTest.cs

@@ -0,0 +1,57 @@
+//
+// 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 hyperbolic cosecant beside a power of a binomial in an even power of the sine, where
+    /// <c>csch(y)</c> arrives as the difference of the exponentials with its half gathered
+    /// elsewhere: Rubi's 6.1.7 <c>csch(c + d x) (a + b sinh(c + d x)^4)^n</c>, which was left
+    /// unevaluated.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnOddHyperbolicIntegrandWithAPowerOfABinomialIntegralTest
+    {
+        private static void DifferentiatesBack(string integrand, params (string, double)[] pins)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+
+            var pinned = integral.Substitute("C", 0);
+            Entity original = integrand.ToEntity();
+            foreach (var (name, value) in pins)
+            {
+                pinned = pinned.Substitute(name, value);
+                original = original.Substitute(name, value);
+            }
+            var derivative = pinned.Differentiate("x");
+            foreach (var at in new[] { -2.3, -0.7, 0.4, 1.9 })
+            {
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                var want = original.Substitute("x", at).EvalNumerical();
+                var difference = Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart);
+                var scale = Math.Max(1.0, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart));
+                Assert.True(difference / scale < 1e-8,
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+        }
+
+        [Theory]
+        [InlineData("csch(c+d*x)*(a+b*sinh(c+d*x)^4)^2")]
+        [InlineData("csch(c+d*x)*(a+b*sinh(c+d*x)^4)^3")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("a", 1.3), ("b", 0.7), ("c", 0.3), ("d", 1.1));
+            DifferentiatesBack(integrand, ("a", -0.5), ("b", 1.5), ("c", -0.4), ("d", 0.8));
+        }
+    }
+}