Flatten a whole power of a constant times an exponential

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

@@ -22156,17 +22156,35 @@ private static (Entity Constant, Entity Base, Entity ExponentInU, Variable U, En
         /// exponential of the product: <c>(e^x)^(1/3)</c> is <c>e^(x/3)</c> and <c>(3^(3x))^(1/4)</c>
         /// is <c>3^(3x/4)</c>, exactly, since the base is positive; and it is only in that
         /// spelling that the exponential rules read them. Timofeev's
-        /// <c>(cos(x/2) + sin(x/2))/(e^x)^(1/3)</c> and <c>cos(3x/2)/(3^(3x))^(1/4)</c>.
+        /// <c>(cos(x/2) + sin(x/2))/(e^x)^(1/3)</c> and <c>cos(3x/2)/(3^(3x))^(1/4)</c>. And a whole
+        /// power of a constant times one, <c>(-e^(-b x))^2</c>, as the constant's power times the
+        /// exponential of the multiple, whatever the constant is.
         /// </summary>
         internal static Entity? SolveByFlatteningAPowerOfAnExponential(Entity expr, Entity.Variable x, bool integrateByParts)
         {
-            var flattened = expr.Replace(node =>
-                node is Powf(Powf(var @base, var inner), var outer)
-                && !@base.ContainsNode(x) && inner.ContainsNode(x) && !outer.ContainsNode(x)
-                && (@base == MathS.e || @base.Evaled is Number.Real { IsPositive: true })
-                && outer.Evaled is Number.Real
-                    ? MathS.Pow(@base, (outer * inner).InnerSimplified)
-                    : node);
+            bool IsAnExponential(Entity @base, Entity inner) =>
+                !@base.ContainsNode(x) && inner.ContainsNode(x)
+                && (@base == MathS.e || @base.Evaled is Number.Real { IsPositive: true });
+            // A whole power of a constant times an exponential: (k E)^n is k^n E^n for every k and
+            // E when n is whole, so (-e^(-b x))^2 is e^(-2 b x). The product of two of them, which
+            // sinh(b x)^3 sinh(d x)^3 expands into, is read by nothing else.
+            Entity? Distributed(Entity left, Entity right, Number.Integer whole) =>
+                (left, right) switch
+                {
+                    (Powf(var b, var p), var k) when !k.ContainsNode(x) && IsAnExponential(b, p) =>
+                        MathS.Pow(k, whole) * MathS.Pow(b, (whole * p).InnerSimplified),
+                    (var k, Powf(var b, var p)) when !k.ContainsNode(x) && IsAnExponential(b, p) =>
+                        MathS.Pow(k, whole) * MathS.Pow(b, (whole * p).InnerSimplified),
+                    _ => null
+                };
+            var flattened = expr.Replace(node => node switch
+            {
+                Powf(Powf(var @base, var inner), var outer)
+                    when IsAnExponential(@base, inner) && !outer.ContainsNode(x) && outer.Evaled is Number.Real =>
+                    MathS.Pow(@base, (outer * inner).InnerSimplified),
+                Powf(Mulf(var left, var right), Number.Integer whole) => Distributed(left, right, whole) ?? node,
+                _ => node
+            });
             // The same question in another spelling, asked as one of its own, so that the
             // closed rules that answer only at the top -- the exponential times a
             // trigonometric -- are consulted for it.

Sources/Tests/UnitTests/Calculus/AWholePowerOfAMultipleOfAnExponentialIntegralTest.cs

@@ -0,0 +1,54 @@
+//
+// 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 whole power of a constant times an exponential, <c>(k e^u)^n = k^n e^(n u)</c>. The terms of
+    /// <c>sinh(a + b x)^3 sinh(c + d x)^3</c>, written in exponentials, are products of two of them,
+    /// <c>(-e^(-b x))^2 (-e^(-d x))^2</c>, which nothing else read: 2.5.0 answered the product, and
+    /// 2.6.0's development had stopped. Rubi's 6.7.1.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AWholePowerOfAMultipleOfAnExponentialIntegralTest
+    {
+        [Theory]
+        [InlineData("(-e^(-b*x))^2*(-e^(-d*x))^2")]
+        [InlineData("(-e^(-b*x))^2")]
+        [InlineData("(2*e^x)^3")]
+        [InlineData("sinh(b*x)^3*sinh(d*x)^3")]
+        [InlineData("sinh(a + b*x)^3*sinh(c + d*x)^3")]
+        public void AsAnExponentialOfTheMultiple(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[] { -1.2, -0.7, 0.3, 0.8, 1.3, 2.9 })
+            {
+                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}");
+        }
+    }
+}