A cos + i A sin below the bar is written as the exponential it is

BREAKING-CHANGES.md

@@ -1371,6 +1371,26 @@ and those rows want partial fractions over the two linear factors instead.
 
 Rubi's 4.7.2: 215 to 237 of 290, no row lost, 20 timeouts to 16.
 
+### `A cos + i A sin` below the bar is written as the exponential it is
+
+`cos(x)^2/(a cos(x) + i a sin(x))^3` was declined after twelve seconds. `A cos(y) + i A sin(y)` is
+`A e^(i y)`, and `A cos(y) - i A sin(y)` is `A e^(-i y)`; written so, the integrand is an
+exponential times a power of a cosine, which the closed rules answer at once. As a sum of a
+cosine and a sine it has no real rotation -- `A^2 + (i A)^2` is zero -- so the rule for
+`a cos + b sin` has nothing to do with it. A whole power of the product written is split,
+`(A e^(i y))^n` as `A^n e^(i n y)`, since the rule that distributes such powers takes a symbolic
+constant and leaves a number: `cos(x)^2/(2 e^(i x))^3` was declined where
+`cos(x)^2/(a e^(i x))^3` was not.
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"cos(x)^2/(a*cos(x)+i*a*sin(x))^3".Integrate("x")` | left unevaluated | exponentials of `-3 i x` times `cos(2x)`, `sin(2x)` |
+| `"1/(a*cos(x)+i*a*sin(x))^3".Integrate("x")` | left unevaluated | `a^(-3) e^(-3 i x)/(-3 i)` |
+
+A secant above the bar is still declined: `sec(x)^2 e^(-3 i x)` is rational in `e^(i x)` with an
+imaginary rate, which nothing answers yet. Rubi's 4.7.2: 237 to 249 of 290, no row lost, 16
+timeouts to 14.
+
 ### `binomial(n, k)` is a function
 
 **Addition, not silent.** The binomial coefficient is a node, `Entity.Binomialf`, spelled

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

@@ -15432,6 +15432,54 @@ static bool IsAWholeNumberOfSteps(ERational difference)
             return rewritten == below ? null : Integration.ComputeAsTheSameQuestion((above / rewritten).InnerSimplified, x, integrateByParts);
         }
 
+        /// <summary>
+        /// <c>A cos(y) + i A sin(y)</c> below the bar, written as the exponential it is:
+        /// <c>A e^(i y)</c>, and <c>A cos(y) - i A sin(y)</c> as <c>A e^(-i y)</c>. Beside a power
+        /// of the cosine above the bar that is an exponential times a power of a cosine, which
+        /// the closed rules answer at once; as written it is a sum of a cosine and a sine whose
+        /// coefficients have no real rotation -- <c>A^2 + (i A)^2</c> is zero -- and
+        /// <c>cos(x)^2/(a cos(x) + i a sin(x))^3</c> was declined after twelve seconds.
+        /// </summary>
+        /// <remarks>
+        /// The sibling of <see cref="SolveByWritingAnImaginaryTangentAsAnExponential"/>, which reads
+        /// <c>A + i A tan(y)</c>; below the bar only, for the same reason.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </remarks>
+        internal static Entity? SolveByWritingAnImaginarySumOfACosineAndASineAsAnExponential(Entity expr, Entity.Variable x, bool integrateByParts)
+        {
+            var (above, below) = Functions.SingleQuotient.Of(Functions.SingleQuotient.Combine(expr));
+            if (!below.ContainsNode(x))
+                return 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))
+                    return node;
+                // The ratio is the imaginary unit one way or the other, and nothing else. Bare:
+                // `i a/a` simplifies to `i provided not a = 0`, and a condition is not a number
+                // to compare against.
+                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;
+                return cosineCoefficient * MathS.Pow(MathS.e, ((plus ? MathS.i : -MathS.i) * argument).InnerSimplified);
+            });
+            if (rewritten == below)
+                return null;
+            // And a whole power of what was written split, `(A e^(i y))^n` as `A^n e^(i n y)`, so
+            // that the exponential stands on its own for the rules that read one: the rule that
+            // distributes such powers takes a constant with a symbol in it and leaves a number,
+            // and `cos(x)^2/(2 e^(i x))^3` was declined where `cos(x)^2/(a e^(i x))^3` was not.
+            rewritten = rewritten.Replace(node =>
+                node is Powf(Mulf(var left, var right), Number.Integer power)
+                && (left is Powf(var leftBase, _) && leftBase == MathS.e || right is Powf(var rightBase, _) && rightBase == MathS.e)
+                    ? 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

@@ -504,6 +504,8 @@ private static Entity Normalized(Entity expr, Entity.Variable x) =>
             // `A + i A tan(z)` is `A e^(i z)/cos(z)`, which beside a polynomial is a shape the
             // closed rules answer, where the imaginary unit in the coefficient is read by none.
             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;
             // A fractional power of a perfect square is the power of the modulus, sgn(P) P^(2r).
             if ((answer = IndefiniteIntegralSolver.SolveByTakingARootOfAPerfectSquare(expr, x, integrateByParts)) is { }) return answer;
             // x^(n - 1) g(x^n) with a symbolic n is g(u)/n under u = x^n.

Sources/Tests/UnitTests/Calculus/PolynomialExponentialTrigTest.cs

@@ -1,4 +1,4 @@
-//
+//
 // Copyright (c) 2019-2026 Angouri.
 // AngouriMath is licensed under MIT.
 // Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
@@ -263,6 +263,37 @@ public void TheResonanceIsSeenSymbolically(string integrand, string? symbol)
         [InlineData("(1 + 2*x)/(3 + 3*i*tan(1/2 + x))^2")]
         public void AnImaginaryTangentBelowTheBarIsAnExponential(string integrand) => DifferentiatesBack(integrand);
 
+        /// <summary>
+        /// And its sibling, <c>A cos(y) + i A sin(y)</c>, which is <c>A e^(i y)</c>: a sum of a
+        /// cosine and a sine with no real rotation, since <c>A^2 + (i A)^2</c> is zero, and beside
+        /// a power of the cosine an exponential times that power once written so. Rubi's 4.7.2,
+        /// fourteen rows; <c>cos(x)^2/(a cos(x) + i a sin(x))^3</c> was declined after twelve
+        /// seconds.
+        /// </summary>
+        [Theory]
+        [InlineData("cos(x)^2/(2*cos(x) + 2*i*sin(x))^3")]
+        [InlineData("cos(x)^5/(2*cos(x) + 2*i*sin(x))^2")]
+        [InlineData("1/(3*cos(1/2 + x) + 3*i*sin(1/2 + x))^3")]
+        [InlineData("cos(x)/(2*cos(x) - 2*i*sin(x))")]
+        public void AnImaginarySumOfACosineAndASineBelowTheBarIsAnExponential(string integrand) => DifferentiatesBack(integrand);
+
+        /// <summary>The same with the coefficient a symbol, pinned only after integrating.</summary>
+        [Fact]
+        public void AnImaginarySumWithASymbolicCoefficient()
+        {
+            var integral = "cos(x)^2/(a*cos(x) + i*a*sin(x))^3".ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            var derivative = integral.Substitute("C", 0).Substitute("a", 0.9).Differentiate("x");
+            var original = "cos(x)^2/(a*cos(x) + i*a*sin(x))^3".ToEntity().Substitute("a", 0.9);
+            foreach (var at in Points)
+            {
+                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);
+                Assert.True(difference < 1e-9, $"at x = {at}: {got} for {want}");
+            }
+        }
+
         /// <summary>
         /// A base with symbols in it, and an exponent whose constant part has them too:
         /// <c>F^(c (a + b x))</c> leaves <c>F^(a c)</c> as a constant factor, and that factor