Integrate a power of x times a hyperbolic function of a power under that power

BREAKING-CHANGES.md

@@ -5888,6 +5888,27 @@ exponential over the derivative of its sine or cosine is now asked as a rational
 | `"exp(n*sinh(a + b*x))*sinh(2*(a + b*x))".ToEntity().Integrate("x")` | `integral(...)` | elementary in `sinh(a + b x)` and its exponential |
 | `"exp(n*cosh(1/2*(a + b*x)))*sinh(a + b*x)".ToEntity().Integrate("x")` | `integral(...)` | the same in `cosh((a + b x)/2)` |
 
+### A power of x times a hyperbolic function of a power of x is integrated under that power
+
+**Answers where there were none.** `x sech(a + b x^2)^7`, `x/(a + b sech(c + d x^2))^2` and their cosecant twins, Rubi's
+6.5.2 and 6.6.2, `sinh(a + b x^2)^3/x^3` and its cosine, 6.1.3 and 6.2.3, and `x^5 cosh(a + b x^3)^7 sinh(a + b x^3)`,
+6.7.1, were declined. The library writes the hyperbolic functions as exponentials, and the rules before this one spent
+the budget on a power of a quotient of exponentials of `a + b x^2`. With `x^m` beside a rational function of hyperbolic
+functions of `c + d x^n`, `n` at least 2 and dividing `m + 1`, the integrand is now asked in `t = x^n`, where
+`x^m dx` is `t^(k - 1) dt/n` for `k = (m + 1)/n`, exactly wherever it is defined since the powers are whole. Only with
+a whole power past the first of a sum or a quotient of the exponentials among them: `x/(a + b sech(c + d x^2))` and
+`x^3 sinh(a + b x^2)` are answered as written, in fewer characters
+([#233](https://github.com/asc-community/AngouriMath/issues/233)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x*sech(a + b*x^2)^7".ToEntity().Integrate("x")` | `integral(...)` | 494 characters |
+| `"x/(a + b*sech(c + d*x^2))^2".ToEntity().Integrate("x")` | `integral(...)` | 2,193 characters |
+| `"sinh(a + b*x^2)^3/x^3".ToEntity().Integrate("x")` | `integral(...)` | 480 characters |
+| `"x^5*cosh(a + b*x^3)^7*sinh(a + b*x^3)".ToEntity().Integrate("x")` | `integral(...)` | 2,824 characters |
+| `"x*(a + b*sech(c + d*x^2))^2".ToEntity().Integrate("x")` | `integral(...)`; 205 characters, in 8.9 s, on the unreleased master | 185 characters, in 45 ms |
+| `"x/(cosh(a + b*x^2)^2 + 1)".ToEntity().Integrate("x")` | `integral(...)`; 145 characters, in 8.3 s, on the unreleased master | 231 characters, in 76 ms |
+
 ### A hyperbolic function of a logarithm is integrated, the exponent folded structurally
 
 `tanh(ln(x))` was left as written. The library spells `tanh(y)` with `e^(2y)`, so a hyperbolic

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

@@ -27796,6 +27796,109 @@ when two.Equals(EInteger.FromInt32(2)) && e1 == MathS.e && e2 == MathS.e
             return answer.Nodes.Any(node => node == MathS.NaN) ? null : answer;
         }
 
+        /// <summary>
+        /// A power of x times a rational function of hyperbolic functions of a linear in a whole
+        /// power of x, <c>x^m F(sinh(c + d x^n), cosh(c + d x^n))</c> with <c>n &gt;= 2</c> dividing
+        /// <c>m + 1</c>, under <c>t = x^n</c>: <c>x^m dx</c> is <c>t^(k - 1) dt/n</c> for
+        /// <c>k = (m + 1)/n</c>, and the question in <c>t</c> is the same function of a linear in
+        /// <c>t</c> beside a whole power of <c>t</c>.
+        /// </summary>
+        /// <remarks>
+        /// The library writes the hyperbolic functions as exponentials, so <c>x sech(a + b x^2)^7</c>
+        /// arrives as x beside a seventh power of a quotient of exponentials of <c>a + b x^2</c>, and
+        /// the rules before this one spent the budget on it: Rubi's 6.5.2 and 6.6.2
+        /// <c>x sech(a + b x^2)^7</c>, <c>x/(a + b sech(c + d x^2))^2</c> and the two with the
+        /// cosecant, 6.1.3 and 6.2.3 <c>sinh(a + b x^2)^3/x^3</c> and its cosine, and 6.7.1
+        /// <c>x^5 cosh(a + b x^3)^7 sinh(a + b x^3)</c>, each of which in <c>t</c> is answered in
+        /// under a second. Exact wherever the integrand is defined, since the powers are whole:
+        /// <c>x^m</c> is <c>x^(n - 1) (x^n)^(k - 1)</c> at every <c>x</c> but 0. Exponentials of a
+        /// linear in <c>x^n</c> and whole powers of them only, with a whole power past the first of
+        /// a sum or a quotient of them among them: <c>x/(a + b sech(c + d x^2))</c> and
+        /// <c>x^3 sinh(a + b x^2)</c> are answered as written, in fewer characters than in
+        /// <c>t</c>, and are left to that. After the rules that answer a polynomial beside
+        /// exponentials of a quadratic as written, which it leaves as they were; a symbolic
+        /// <c>n</c> is <see cref="SolveByAPowerOfTheVariableTimesAFunctionOfItsPower"/>'s.
+        /// https://github.com/asc-community/AngouriMath/issues/233
+        /// </remarks>
+        internal static Entity? SolveAPowerOfXTimesAHyperbolicFunctionOfAPowerUnderThatPower(Entity expr, Entity.Variable x, bool integrateByParts)
+        {
+            if (!Integration.AnsweringTheQuestionAsked)
+                return null;
+            var m = 0;
+            Entity constant = Number.Integer.One;
+            Entity? above = null, below = null;
+            foreach (var (factor, underneath) in FactorsOfTheIntegrand(expr))
+            {
+                if (!factor.ContainsNode(x))
+                    constant = underneath ? constant / factor : constant * factor;
+                else if (factor == x)
+                    m += underneath ? -1 : 1;
+                else if (factor is Powf(var b0, Number.Integer { EInteger: var p0 }) && b0 == x && p0.CanFitInInt32())
+                    m += underneath ? -p0.ToInt32Unchecked() : p0.ToInt32Unchecked();
+                else if (underneath)
+                    below = below is null ? factor : below * factor;
+                else
+                    above = above is null ? factor : above * factor;
+            }
+            if (above is null && below is null)
+                return null;
+            var rest = below is null ? above! : (above ?? Number.Integer.One) / below;
+            // The one whole power of x the rest is written in.
+            var powers = rest.Nodes.Where(node => node is Powf(var b1, Number.Integer) && b1 == x).Distinct().ToList();
+            if (powers.Count != 1 || powers[0] is not Powf(_, Number.Integer { EInteger: var power }) || !power.CanFitInInt32())
+                return null;
+            var n = power.ToInt32Unchecked();
+            if (n < 2 || n > 12 || (m + 1) % n != 0)
+                return null;
+            var t = Variable.CreateUnique(expr, "t_hyp");
+            var inT = rest.Substitute(powers[0], t);
+            if (inT.ContainsNode(x))
+                return null;
+            // Exponentials of a linear in t with a constant base, and whole powers, sums,
+            // products and quotients of them: nothing else of t. With a whole power past the
+            // first of a sum of them, or of a quotient: a hyperbolic function to the first power
+            // is answered in x as written, by the rules before this one, and in fewer characters.
+            var anExponential = false;
+            var aWholePower = false;
+            foreach (var node in inT.Nodes)
+            {
+                if (!node.ContainsNode(t))
+                    continue;
+                switch (node)
+                {
+                    case Variable or Sumf or Minusf or Mulf or Divf:
+                        continue;
+                    case Powf(var sum, Number.Integer { EInteger: var whole }):
+                        aWholePower |= sum is Sumf or Minusf or Mulf or Divf && whole.Abs().CompareTo(EInteger.One) > 0;
+                        continue;
+                    case Powf(var @base, var exponent) when !@base.ContainsNode(t)
+                        && TreeAnalyzer.TryGetPolyLinear(exponent, t, out var slope, out _) && !slope.ContainsNode(t)
+                        && !Functions.PartialFractions.IsZeroAsAValue(slope):
+                        anExponential = true;
+                        continue;
+                    default:
+                        return null;
+                }
+            }
+            if (!anExponential || !aWholePower)
+                return null;
+            // A bare t outside every exponential is a power of x that n does not reach.
+            if (inT.Replace(node => node is Powf(var b2, var e2) && !b2.ContainsNode(t) && e2.ContainsNode(t) ? Number.Integer.One : node).ContainsNode(t))
+                return null;
+            var k = (m + 1) / n;
+            var question = k switch
+            {
+                1 => inT,
+                > 1 => MathS.Pow(t, Number.Integer.Create(k - 1)) * inT,
+                _ => inT / MathS.Pow(t, Number.Integer.Create(1 - k)),
+            };
+            if (Integration.ComputeAsAQuestionOfItsOwn(question, t, integrateByParts) is not { } integrated
+                || integrated.Nodes.Any(node => node is Integralf))
+                return null;
+            var answer = (constant / Number.Integer.Create(n) * integrated.Substitute(t, powers[0])).InnerSimplified;
+            return answer.Nodes.Any(node => node == MathS.NaN) ? null : answer;
+        }
+
         /// <summary>
         /// A product of whole powers of a sine and a cosine of one linear times something else in x
         /// that is no polynomial, <c>sin(u)^3 cos(u)^6/(a + b u)</c>, linearised: the product is a sum

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

@@ -877,6 +877,10 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             if ((answer = IndefiniteIntegralSolver.SolveByTheDerivativeOfAQuadraticAsTheVariable(expr, x, integrateByParts)) is { }) return answer;
             // x^(n - 1) g(x^n) with a symbolic n is g(u)/n under u = x^n.
             if ((answer = IndefiniteIntegralSolver.SolveByAPowerOfTheVariableTimesAFunctionOfItsPower(expr, x, integrateByParts)) is { }) return answer;
+            // And x^(k n - 1) g(x^n) with a whole n >= 2, for a whole power of hyperbolic functions of
+            // a linear in x^n, which arrive as exponentials: `x sech(a + b x^2)^7` is
+            // `sech(a + b t)^7/2` under t = x^2, where the rules before this spent the budget.
+            if ((answer = IndefiniteIntegralSolver.SolveAPowerOfXTimesAHyperbolicFunctionOfAPowerUnderThatPower(expr, x, integrateByParts)) is { }) return answer;
             // A whole power of a product of a constant and the variable, as the product of
             // the powers, which is how the inverse hyperbolic secant and cosecant arrive.
             if ((answer = IndefiniteIntegralSolver.SolveByDistributingWholePowersOfProducts(expr, x, integrateByParts)) is { }) return answer;

Sources/Tests/UnitTests/Calculus/APowerOfXTimesAHyperbolicFunctionOfAPowerIntegralTest.cs

@@ -0,0 +1,59 @@
+//
+// 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 x times a whole power of hyperbolic functions of a linear in <c>x^n</c>, under
+    /// <c>t = x^n</c>, where <c>x^m dx</c> is <c>t^((m + 1)/n - 1) dt/n</c>. The hyperbolic functions
+    /// arrive as exponentials of <c>a + b x^n</c>, and in x the rules spent the budget on each of
+    /// these. Rubi's 6.1.3, 6.2.3, 6.5.2, 6.6.2 and 6.7.1.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class APowerOfXTimesAHyperbolicFunctionOfAPowerIntegralTest
+    {
+        [Theory]
+        [InlineData("sinh(a + b*x^2)^3/x^3")]
+        [InlineData("cosh(a + b*x^2)^3/x^3")]
+        [InlineData("x*sech(a + b*x^2)^7")]
+        [InlineData("x*csch(a + b*x^2)^7")]
+        [InlineData("x/(a + b*sech(c + d*x^2))^2")]
+        [InlineData("x/(a + b*csch(c + d*x^2))^2")]
+        [InlineData("x*(a + b*sech(c + d*x^2))^2")]
+        [InlineData("x^5*cosh(a + b*x^3)^7*sinh(a + b*x^3)")]
+        [InlineData("sinh(a + b*x^2)^4/x^5")]
+        [InlineData("x/(cosh(a + b*x^2)^2 + 1)")]
+        public void UnderThePower(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 5000, $"{text.Length} characters of answer for {integrand}");
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.4).Substitute("d", 0.3);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -1.3, -0.8, -0.4, 0.35, 0.9, 1.4 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                if (want.IsNaN)
+                    continue;
+                compared++;
+                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}");
+            }
+            Assert.True(compared >= 5, $"only {compared} points could be compared for {integrand}");
+        }
+    }
+}