Integrate a symbolic power of a trigonometric function where it is the derivative of a product

BREAKING-CHANGES.md

@@ -396,6 +396,20 @@ in Rubi's 1.1.3.4 come out shorter as well, and those that took two seconds take
 | `"1/(x^8*(a + b*x^3)*(c + d*x^3))".ToEntity().Integrate("x")` | `integral(...)` | 1,217 characters |
 | `"1/(x^6*(a + b*x^4)*(c + d*x^4))".ToEntity().Integrate("x")` | `integral(...)` | 795 characters |
 
+### A symbolic power of a sine, cosine, secant or cosecant is integrated where it is the derivative of a product
+
+**Answers where there were none.** `sec(c + d x)^m (m - (1 + m) sec(c + d x)^2)` was declined, and so were five more
+of Rubi's 4.1.4.1, 4.2.4.1 and 4.5.4.1. `f^m (alpha + beta f^2)` for `f` a sine, cosine, secant or cosecant of a linear
+is the derivative of `cos sin^(m + 1)`, `sin cos^(m + 1)`, `tan sec^m` or `cot csc^m` where its bracket is a multiple of
+that derivative's, and is now integrated so ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sec(c+d*x)^m*(-q*m/(1+m)+q*sec(c+d*x)^2)".ToEntity().Integrate("x")` | `integral(...)` | 75 characters |
+| `"sec(c+d*x)^m*(A-A*(1+m)*sec(c+d*x)^2/m)".ToEntity().Integrate("x")` | `integral(...)` | 54 characters |
+| `"sec(k+f*x)^m*(m-(1+m)*sec(k+f*x)^2)".ToEntity().Integrate("x")` | `integral(...)` | 54 characters |
+| `"sin(k+f*x)^m*(1+m-(2+m)*sin(k+f*x)^2)".ToEntity().Integrate("x")` | `integral(...)` | 71 characters |
+
 ### A power of a sine or hyperbolic sine over a binomial in a power of it is split over the binomial's roots
 
 **Answers where there were none.** `sinh(c + d x)^4/(a + b sinh(c + d x)^3)` was declined, and so were eight more of

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

@@ -29317,6 +29317,53 @@ bool HoldsItsValue(Entity factor)
             return (s * logarithm - remainder).InnerSimplified;
         }
 
+        /// <summary>
+        /// <c>f^m (alpha + beta f^2)</c> for a symbolic <c>m</c> and <c>f</c> a sine, cosine, secant or
+        /// cosecant of a linear <c>u</c>, where it is the derivative of a product: with <c>u' = 1</c>,
+        /// <c>(cos sin^(m + 1))' = sin^m ((m + 1) - (m + 2) sin^2)</c>,
+        /// <c>(sin cos^(m + 1))' = cos^m ((m + 2) cos^2 - (m + 1))</c>,
+        /// <c>(tan sec^m)' = sec^m ((m + 1) sec^2 - m)</c> and <c>(cot csc^m)' = -csc^m ((m + 1) csc^2 - m)</c>.
+        /// </summary>
+        /// <remarks>
+        /// The integrand is read as <c>f^m</c> times a polynomial in <c>f</c> with only the constant and
+        /// square terms, <c>alpha + beta f^2</c>, which matches <c>p0 + p2 f^2</c> above where
+        /// <c>beta p0 = alpha p2</c>; the answer is <c>(alpha/p0) G/u'</c>. Rubi's 4.1.4.1, 4.2.4.1 and 4.5.4.1
+        /// <c>sec(c + d x)^m (m - (1 + m) sec(c + d x)^2)</c> and the like were declined.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </remarks>
+        internal static Entity? SolveASymbolicPowerOfATrigonometricFunctionAsTheDerivativeOfAProduct(Entity expr, Entity.Variable x)
+        {
+            if (!Integration.AnsweringTheQuestionAsked)
+                return null;
+            // The power with a symbol in its exponent, of a sine, cosine, secant or cosecant of a linear.
+            var power = expr.Nodes.OfType<Powf>().FirstOrDefault(node => node.Exponent.Vars.Any() && !node.Exponent.ContainsNode(x)
+                && node.Base is Sinf or Cosf or Secantf or Cosecantf);
+            if (power is not Powf(var function, var m))
+                return null;
+            var argument = function switch { Sinf(var y) => y, Cosf(var y) => y, Secantf(var y) => y, Cosecantf(var y) => y, _ => null };
+            if (argument is null || !TreeAnalyzer.TryGetPolyLinear(argument, x, out var slope, out _) || Functions.PartialFractions.IsZeroAsAValue(slope))
+                return null;
+            var s = Variable.CreateUnique(expr, "s_trig");
+            // The power taken off by writing 1 for it: `s^m (...)/s^m` is not cancelled by the simplification.
+            var inS = expr.Substitute(power, Number.Integer.One).Substitute(function, s).InnerSimplified;
+            if (inS.ContainsNode(x) || !TreeAnalyzer.TryGetPolynomial(inS.Expand(), s, out var read)
+                || read.Keys.Any(key => !key.IsZero && !key.Equals(EInteger.FromInt32(2))))
+                return null;
+            var alpha = read.TryGetValue(EInteger.Zero, out var a0) ? a0 : Number.Integer.Zero;
+            var beta = read.TryGetValue(EInteger.FromInt32(2), out var b2) ? b2 : Number.Integer.Zero;
+            var (p0, p2, product) = function switch
+            {
+                Sinf => (m + 1, -(m + 2), MathS.Cos(argument) * MathS.Pow(function, m + 1)),
+                Cosf => (-(m + 1), m + 2, MathS.Sin(argument) * MathS.Pow(function, m + 1)),
+                Secantf => (-m, m + 1, MathS.Tan(argument) * MathS.Pow(function, m)),
+                _ => (m, -(m + 1), MathS.Cotan(argument) * MathS.Pow(function, m)),
+            };
+            if (Functions.PartialFractions.IsZeroAsAValue(p0.InnerSimplified)
+                || !Functions.PartialFractions.IsZeroAsAValue((beta * p0 - alpha * p2).Expand().InnerSimplified))
+                return null;
+            return (alpha / (p0 * slope) * product).InnerSimplified;
+        }
+
         /// <summary>
         /// A power of a constant multiple of a radicand that a logarithm holds under a root,
         /// written over that radicand: <c>acosh(c x)</c> is <c>ln(c x + sqrt(c^2 x^2 - 1))</c>,

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

@@ -664,6 +664,9 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // A sum free of the variable that comes to a number multiplied out, as that number,
             // before anything reads a coefficient or an exponent in it: `2 + 2 a - 2 (1 + a)`.
             if ((answer = IndefiniteIntegralSolver.SolveByFoldingConstantSumsToTheirNumbers(expr, x, integrateByParts)) is { }) return answer;
+            // A symbolic power of a sine, cosine, secant or cosecant times `alpha + beta f^2`, where it is
+            // the derivative of `cos sin^(m + 1)`, `sin cos^(m + 1)`, `tan sec^m` or `cot csc^m`.
+            if ((answer = IndefiniteIntegralSolver.SolveASymbolicPowerOfATrigonometricFunctionAsTheDerivativeOfAProduct(expr, x)) is { }) return answer;
             // A power of a sine, cosine or hyperbolic one over a binomial in a power of it, over the
             // binomial's roots, before the rules that spent the budget on `sinh(y)^4/(a + b sinh(y)^3)`.
             if ((answer = IndefiniteIntegralSolver.SolveAPowerOverABinomialInAPowerOfASineOverItsRoots(expr, x)) is { }) return answer;

Sources/Tests/UnitTests/Calculus/ASymbolicPowerOfATrigonometricFunctionAsADerivativeIntegralTest.cs

@@ -0,0 +1,69 @@
+//
+// 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 symbolic power of a sine, cosine, secant or cosecant times <c>alpha + beta f^2</c>, where it is
+    /// the derivative of <c>cos sin^(m + 1)</c>, <c>sin cos^(m + 1)</c>, <c>tan sec^m</c> or <c>cot csc^m</c>:
+    /// Rubi's 4.1.4.1 and 4.5.4.1.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ASymbolicPowerOfATrigonometricFunctionAsADerivativeIntegralTest
+    {
+        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}");
+            }
+        }
+
+        // d with the sign that keeps the argument real at every sampled x, and |d| below 2/2.3^2
+        // so that the inverse sine and cosine stay inside [-1, 1]; off it the exact checks hold too.
+        [Theory]
+        [InlineData("sec(c+d*x)^m*(-q*m/(1+m)+q*sec(c+d*x)^2)")]
+        [InlineData("sec(c+d*x)^m*(A-A*(1+m)*sec(c+d*x)^2/m)")]
+        [InlineData("sin(c+d*x)^m*(1+m-(2+m)*sin(c+d*x)^2)")]
+        [InlineData("cos(x)^m*(-(1+m)+(2+m)*cos(x)^2)")]
+        [InlineData("csc(x)^m*(m-(1+m)*csc(x)^2)")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("A", 0.6), ("q", 1.25), ("c", 0.3), ("d", 0.75), ("m", 2.5));
+            DifferentiatesBack(integrand, ("A", -0.75), ("q", 0.4), ("c", -1), ("d", -0.5), ("m", 1.0 / 3));
+        }
+
+        [Fact]
+        public void IsNotClaimedWhereTheBracketIsNotTheDerivative()
+        {
+            // beta p0 != alpha p2: (1 + m) sec^2 - (m + 1) is not the derivative of tan sec^m.
+            var integral = "sec(x)^m*(1+m-(1+m)*sec(x)^2)".ToEntity().Integrate("x").Stringize();
+            Assert.DoesNotContain("tan(x) * sec(x) ^ m", integral);
+        }
+    }
+}