An exponential of the sine or the cosine beside a polynomial in x, the sine and the cosine is integrated

BREAKING-CHANGES.md

@@ -568,6 +568,20 @@ symbol in it and a denominator past a sextic is now left to the partial fraction
 | `"(d + e*x^2 + f*x^4)/(x^4*(a + b*x^2 + c*x^4)^2)".ToEntity().Integrate("x")` | `integral(...)`; past a minute on the unreleased master | 3,715 characters |
 | `"cos(e + f*x)^5/(a + b*sec(e + f*x)^2)^3".ToEntity().Integrate("x")` | `integral(...)`; past a minute on the unreleased master | 27,944 characters |
 
+### An exponential of the sine or the cosine beside a polynomial in `x`, the sine and the cosine is integrated
+
+**Answers where there were none.** `e^sin(x) sec(x)^2 (x cos(x)^3 - sin(x))` was declined, Rubi's 4.7.7, and so were
+`e^cos(x) (x sin(x) - 1)` and `e^sin(x) (x cos(x) + 1)`. The ansatz for an exponential beside the trigonometric
+functions read the exponent only when it was linear in `x`; it now also takes an exponent that is a polynomial in
+the sine and the cosine, whose derivative joins each term the ansatz differentiates
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"e^sin(x)*sec(x)^2*(x*cos(x)^3 - sin(x))".ToEntity().Integrate("x")` | `integral(...)` | `e ^ sin(x) * (-1 + x * cos(x)) / cos(x)` |
+| `"e^cos(x)*(x*sin(x) - 1)".ToEntity().Integrate("x")` | `integral(...)` | `e ^ cos(x) * -x` |
+| `"e^sin(x)*(x*cos(x) + 1)".ToEntity().Integrate("x")` | `integral(...)` | `e ^ sin(x) * x` |
+
 ### A symbolic quartic in `x^2` with a square discriminant is split into its two quadratics
 
 **Answers where there were none.** `cot(c + d x)^3 sqrt(a + b sec(c + d x))` ran past a minute: the root of the

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

@@ -14906,8 +14906,9 @@ bool Read(Entity e, int index, int[] prefix)
 
         /// <summary>
         /// A rational function of <c>x</c> and of <c>sin(x)</c> and <c>cos(x)</c> together, with
-        /// an exponential <c>e^(a x)</c> in front or not, closed by the ansatz
-        /// <c>F = e^(a x) P/Q</c> with <c>P</c> and <c>Q</c> polynomials in the three.
+        /// an exponential <c>e^h</c> in front or not, <c>h</c> linear in <c>x</c> or a polynomial in
+        /// the three, closed by the ansatz <c>F = e^h P/Q</c> with <c>P</c> and <c>Q</c> polynomials
+        /// in the three.
         /// </summary>
         /// <remarks>
         /// <para>
@@ -14929,6 +14930,12 @@ bool Read(Entity e, int index, int[] prefix)
         /// tower first, and every substitution of it leaves powers of <c>1 + t^2</c> above and
         /// below that nothing cancels; the ring needs no substitution.
         /// </para>
+        /// <para>
+        /// An exponent in the ring has its derivative there too, so the bracket is
+        /// <c>(h' P Q + P' Q - P Q') D</c> and nothing else changes. Rubi's 4.7.7
+        /// <c>e^sin(x) sec(x)^2 (x cos(x)^3 - sin(x))</c> is <c>(e^sin(x) (x - sec(x)))'</c> so, a sum
+        /// of two terms neither of which has an elementary antiderivative.
+        /// </para>
         /// https://github.com/asc-community/AngouriMath/issues/718
         /// </remarks>
         internal static Entity? SolveByTrigonometricTowerAnsatz(Entity expr, Entity.Variable x)
@@ -14944,11 +14951,15 @@ bool Read(Entity e, int index, int[] prefix)
             if (ReadOneExponentialTimesTheRest(expr, x) is not var (exponent, rest))
                 return null;
             Entity a = Number.Integer.Zero;
+            // Or an exponent that is itself a polynomial in the three, read below: `e^sin(x)`'s
+            // derivative is `cos(x) e^sin(x)`, and the bracket takes `h' m Q` for `a m Q`.
+            var curved = false;
             if (exponent is not null)
             {
-                if (!TreeAnalyzer.TryGetPolyLinear(exponent, x, out var slope, out _) || slope.ContainsNode(x))
-                    return null;
-                a = slope;
+                if (TreeAnalyzer.TryGetPolyLinear(exponent, x, out var slope, out _) && !slope.ContainsNode(x))
+                    a = slope;
+                else
+                    curved = true;
             }
             var overTheTwo = rest.Replace(node => node switch
             {
@@ -14994,6 +15005,23 @@ static bool IsAWholeMultiple(Entity argument, Entity.Variable x, out int k)
             var (above, below) = Functions.SingleQuotient.Of(inTheThree);
             if (!TrigonometricPolynomial.TryRead(above, x, s, c, out var n) || !TrigonometricPolynomial.TryRead(below, x, s, c, out var d))
                 return null;
+            TrigonometricPolynomial? exponentPrime = null;
+            if (curved)
+            {
+                var exponentInTheThree = exponent!.Replace(node =>
+                {
+                    if (node is Sinf(var argument) && IsAWholeMultiple(argument, x, out var k))
+                        return MultipleAngle(k, s, c).Sine;
+                    if (node is Cosf(var argument2) && IsAWholeMultiple(argument2, x, out var k2))
+                        return MultipleAngle(k2, s, c).Cosine;
+                    return node;
+                });
+                if (exponentInTheThree.Nodes.Any(node => node is Sinf or Cosf && node.ContainsNode(x))
+                    || !TrigonometricPolynomial.TryRead(exponentInTheThree, x, s, c, out var h)
+                    || h.DegreeInX > MaximumTrigonometricTowerDegree || h.TrigonometricDegree > MaximumTrigonometricTowerDegree)
+                    return null;
+                exponentPrime = h.Derivative();
+            }
             if (n.DegreeInX > MaximumTrigonometricTowerDegree || d.DegreeInX > MaximumTrigonometricTowerDegree
                 || n.TrigonometricDegree > MaximumTrigonometricTowerDegree || d.TrigonometricDegree > MaximumTrigonometricTowerDegree)
                 return null;
@@ -15037,6 +15065,8 @@ static bool IsAWholeMultiple(Entity argument, Entity.Variable x, out int k)
                             var bracket = m.Derivative().Times(q).Minus(m.Times(qPrime));
                             if (a.Evaled is not Number.Complex { IsZero: true })
                                 bracket = bracket.Plus(m.Times(q).Scaled(a));
+                            if (exponentPrime is not null)
+                                bracket = bracket.Plus(m.Times(q).Times(exponentPrime));
                             columns.Add(bracket.Times(d));
                             monomialsOfP.Add((i, k, flag));
                         }

Sources/Tests/UnitTests/Calculus/AnExponentialOfATrigonometricPolynomialIntegralTest.cs

@@ -0,0 +1,46 @@
+//
+// 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>
+    /// An exponential of a polynomial in <c>x</c>, <c>sin(x)</c> and <c>cos(x)</c> beside a
+    /// rational function of the three, closed by the ansatz <c>e^h P/Q</c>:
+    /// <c>e^sin(x) sec(x)^2 (x cos(x)^3 - sin(x))</c> is <c>(e^sin(x) (x - sec(x)))'</c>, though
+    /// neither of its two terms has an elementary antiderivative. Rubi's 4.7.7.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnExponentialOfATrigonometricPolynomialIntegralTest
+    {
+        [Theory]
+        [InlineData("e^sin(x)*sec(x)^2*(x*cos(x)^3 - sin(x))", 100)]
+        [InlineData("e^cos(x)*(x*sin(x) - 1)", 100)]
+        [InlineData("e^sin(x)*(x*cos(x) + 1)", 100)]
+        public void DifferentiatesBack(string integrand, int atMost)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < atMost, $"{text.Length} characters of answer for {integrand}");
+            var derivative = integral.Substitute("C", 0).Differentiate("x");
+            var original = integrand.ToEntity();
+            foreach (var at in new[] { "0.35", "1.1", "2.3", "4.4" })
+            {
+                var want = original.Substitute("x", at.ToEntity()).EvalNumerical();
+                var got = derivative.Substitute("x", at.ToEntity()).EvalNumerical();
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart)
+                            < 1e-9 * Math.Max(1, Math.Abs((double)want.RealPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+        }
+    }
+}