An imaginary sum of a cosine and a sine beside powers of the secant, over the cosine

BREAKING-CHANGES.md

@@ -381,6 +381,25 @@ two more of Rubi's 4.5.2.3. The two sums multiply to `-a c tan^2`, so in `u = a
 | `"sec(e + f*x)*(a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^(5/2)".ToEntity().Integrate("x")` | `integral(...)` | 469 characters |
 | `"sec(e + f*x)*(a + a*sec(e + f*x))^m*sqrt(c - c*sec(e + f*x))".ToEntity().Integrate("x")` | `integral(...)` | 268 characters |
 
+### `a cos + i a sin` beside powers of the secant is written over the cosine
+
+**Answers where there were none, and shorter ones.** `sec(c + d x)^5/(a cos(c + d x) + i a sin(c + d x))^2` was declined,
+and so were `sec^4` over the cube, and `sec^6` over the sum and over its cube, Rubi's 4.7.2. `A cos(y) ± i A sin(y)` below
+the bar, with nothing but cosines and secants of `y` beside it, is written `A cos(y) (1 ± i tan(y))` now, where the
+exponential it also is gave nothing the rules read: a power of the secant over a power of `1 ± i tan(y)` is what the
+rules for `a + i a tan` take. 2.5.0 declined all six below; where the unreleased master answered, `sec^4` over the sum
+and `sec^2` over its cube, the answers are a quarter and three-fifths the size
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sec(c + d*x)^6/(a*cos(c + d*x) + i*a*sin(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | 419 characters |
+| `"sec(c + d*x)^5/(a*cos(c + d*x) + i*a*sin(c + d*x))^2".ToEntity().Integrate("x")` | `integral(...)` | 668 characters |
+| `"sec(c + d*x)^4/(a*cos(c + d*x) + i*a*sin(c + d*x))^3".ToEntity().Integrate("x")` | `integral(...)` | 639 characters |
+| `"sec(c + d*x)^6/(a*cos(c + d*x) + i*a*sin(c + d*x))^3".ToEntity().Integrate("x")` | `integral(...)` | 771 characters |
+| `"sec(c + d*x)^4/(a*cos(c + d*x) + i*a*sin(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | 209 characters, where the unreleased master gave 780 |
+| `"sec(c + d*x)^2/(a*cos(c + d*x) + i*a*sin(c + d*x))^3".ToEntity().Integrate("x")` | `integral(...)` | 488 characters, where the unreleased master gave 810 |
+
 ### An even rational function with symbols in it is not searched for a substitution
 
 **Answers where there were none.** `(d + e x^2 + f x^4)/(x^4 (a + b x^2 + c x^4)^2)` ran past a minute: the

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

@@ -28604,6 +28604,62 @@ node is Powf(Mulf(var left, var right), Number.Integer power)
             return Integration.ComputeAsTheSameQuestion((above / rewritten).InnerSimplified, x, integrateByParts);
         }
 
+        /// <summary>
+        /// <c>A cos(y) + i A sin(y)</c> below the bar, with nothing but cosines of <c>y</c> and their
+        /// reciprocals beside it, written as <c>A cos(y) (1 + i tan(y))</c>, and <c>A cos(y) - i A sin(y)</c>
+        /// as <c>A cos(y) (1 - i tan(y))</c>: a power of the secant over a power of <c>1 ± i tan(y)</c>,
+        /// which the rules for that sum answer.
+        /// </summary>
+        /// <remarks>
+        /// Where the exponential <see cref="SolveByWritingAnImaginarySumOfACosineAndASineAsAnExponential"/>
+        /// writes declines: an odd power of the secant beside it, `sec(y)^5/(a cos(y) + i a sin(y))^2`,
+        /// is a half-odd power of `1 + tan(y)^2` in the tangent, and that is `(1 - i t)^(5/2) (1 + i t)^(1/2)`
+        /// over the sum, which the rules for `a + i a tan` take. Rubi's 4.7.2.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </remarks>
+        internal static Entity? SolveByWritingAnImaginarySumOfACosineAndASineOverTheCosine(Entity expr, Entity.Variable x, bool integrateByParts)
+        {
+            var (above, below) = Functions.SingleQuotient.Of(Functions.SingleQuotient.Combine(expr));
+            if (!below.ContainsNode(x))
+                return null;
+            Entity? argumentOfThePair = null;
+            var rewritten = below.Replace(node =>
+            {
+                if (node is not (Sumf or Minusf) || !node.ContainsNode(x)
+                    || !TryReadACosineAndASine(node, x, out var cosineCoefficient, out var sineCoefficient, out var argument)
+                    || argumentOfThePair is { } && argumentOfThePair != argument)
+                    return node;
+                var ratio = Functions.PartialFractions.Bare((sineCoefficient / cosineCoefficient).InnerSimplified);
+                if (ratio.Evaled is not Number.Complex)
+                    ratio = Functions.PartialFractions.Bare(ratio.Simplify());
+                var plus = ratio.Evaled == MathS.i.Evaled;
+                if (!plus && ratio.Evaled != (-MathS.i).Evaled)
+                    return node;
+                argumentOfThePair = argument;
+                return cosineCoefficient * MathS.Cos(argument) * (1 + (plus ? MathS.i : -MathS.i) * MathS.Tan(argument));
+            });
+            if (rewritten == below || argumentOfThePair is not { } y)
+                return null;
+            // Nothing beside the pair but cosines and secants of its argument: a sine elsewhere is the
+            // exponential's.
+            bool OnlyCosines(Entity side, bool andTheTangent) => side.Nodes.All(node => node switch
+            {
+                Tanf(var a) => andTheTangent && a == y,
+                Sinf or Tanf or Cotanf or Cosecantf => false,
+                Cosf(var a) => a == y,
+                Secantf(var a) => a == y,
+                _ => true,
+            });
+            if (!OnlyCosines(above, andTheTangent: false) || !OnlyCosines(rewritten, andTheTangent: true))
+                return null;
+            // A whole power of the product split, so that the power of the sum stands on its own.
+            rewritten = rewritten.Replace(node =>
+                node is Powf(Mulf(var left, var right), Number.Integer power) && (left.Nodes.Any(n => n is Tanf) || right.Nodes.Any(n => n is Tanf))
+                    ? MathS.Pow(left, power) * MathS.Pow(right, power)
+                    : node);
+            return Integration.ComputeAsTheSameQuestion((above / rewritten).InnerSimplified, x, integrateByParts);
+        }
+
         /// <summary>
         /// Bioche's first two rules for the hyperbolic functions: a rational function of
         /// <c>sinh(y)</c> and <c>cosh(y)</c> that is odd in the hyperbolic sine is a rational

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

@@ -692,6 +692,7 @@ private static Entity Normalized(Entity expr, Entity.Variable x) =>
             if ((answer = IndefiniteIntegralSolver.SolveByWritingAnImaginaryTangentAsAnExponential(expr, x, integrateByParts)) is { }) return answer;
             // And `A cos(z) + i A sin(z)`, which is `A e^(i z)`, where no rotation is real.
             if ((answer = IndefiniteIntegralSolver.SolveByWritingAnImaginarySumOfACosineAndASineAsAnExponential(expr, x, integrateByParts)) is { }) return answer;
+            if ((answer = IndefiniteIntegralSolver.SolveByWritingAnImaginarySumOfACosineAndASineOverTheCosine(expr, x, integrateByParts)) is { }) return answer;
             // Exponentials of quadratics, with sines, cosines and sums of them multiplied out:
             // each term is the Gaussian's, where the search would take the product by parts.
             if ((answer = IndefiniteIntegralSolver.SolveAProductOfExponentialsOfQuadratics(expr, x)) is { }) return answer;

Sources/Tests/UnitTests/Calculus/AnImaginaryCosineAndSineSumOverTheCosineIntegralTest.cs

@@ -0,0 +1,42 @@
+//
+// 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>
+    /// <c>a cos(y) + i a sin(y)</c> beside powers of the secant is written <c>a cos(y) (1 + i tan(y))</c>,
+    /// which the rules for <c>a + i a tan</c> take. Rubi's 4.7.2.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnImaginaryCosineAndSineSumOverTheCosineIntegralTest
+    {
+        [Theory]
+        [InlineData("sec(x)^5/(a*cos(x) + i*a*sin(x))^2")]
+        [InlineData("sec(x)^6/(a*cos(x) - i*a*sin(x))")]
+        public void DifferentiatesBack(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Entity Pinned(Entity e) => e.Substitute("a", 3);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            foreach (var at in new[] { -1.1, -0.4, 0.3, 1.2 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                var scale = Math.Max(1, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart));
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart) < 1e-9 * scale,
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+        }
+    }
+}