A symbolic power of a + i a tan is gathered with the secant's in the sum

BREAKING-CHANGES.md

@@ -439,9 +439,22 @@ answered without one now ([#718](https://github.com/asc-community/AngouriMath/is
 
 | Input | Was (2.5.0) | Now |
 |---|---|---|
-| `"sec(x)^3*sqrt(a + i*a*tan(x))".ToEntity().Integrate("x")` | `integral(...)` | powers of `sqrt(a - i a tan(x))` times a factor constant wherever it is continuous, 495 characters |
-| `"sec(x)^5/(a + i*a*tan(x))^(3/2)".ToEntity().Integrate("x")` | `integral(...)` | the same, 399 characters |
-| `"cos(x)^9*(a + i*a*tan(x))^(7/2)".ToEntity().Integrate("x")` | `integral(...)`; on the unreleased master, an answer provided `1 + i tan(x) >= 0` | 1,067 characters, unconditional |
+| `"sec(x)^3*sqrt(a + i*a*tan(x))".ToEntity().Integrate("x")` | `integral(...)` | powers of `sqrt(a - i a tan(x))` times a factor constant wherever it is continuous, 285 characters |
+| `"sec(x)^5/(a + i*a*tan(x))^(3/2)".ToEntity().Integrate("x")` | `integral(...)` | the same, 225 characters |
+| `"cos(x)^9*(a + i*a*tan(x))^(7/2)".ToEntity().Integrate("x")` | `integral(...)`; on the unreleased master, an answer provided `1 + i tan(x) >= 0` | 768 characters, unconditional |
+
+### A symbolic power of `a + i a tan` is gathered with the secant's in the sum
+
+**Answers where there were none.** `(k sec(x))^(4 - 2 n) (a + i a tan(x))^n` was declined. In
+`u = a + i a tan(z)` it is `u^n (u/a)^(1 - n) ((2 a - u)/a)^(1 - n)`, whose first two factors are one
+power of `u/a`, and written apart no rule read them. They are gathered now, `u^n` being `a^n (u/a)^n` up
+to a factor constant on the path; the answers to the rest of Rubi's 4.3.1.2 integrated in `u` come out
+shorter by a third or more ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(k*sec(x))^(4 - 2*n)*(a + i*a*tan(x))^n".ToEntity().Integrate("x")` | `integral(...)` | 466 characters |
+| `"(k*sec(x))^(2*n)*(a + i*a*tan(x))^(3 - n)".ToEntity().Integrate("x")` | `integral(...)` | 675 characters |
 
 ### A quotient in `x^2` over a power of a linear in `x^2` and a biquadratic is split in `x^2`
 

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

@@ -27124,8 +27124,13 @@ static Entity SumOfThePowers(Entity first, Entity second)
             }
             var u = Variable.CreateUnique(expr, "u_tan");
             var half = ((secantPower - 2) / 2).InnerSimplified;
-            Entity squared = MathS.Pow(u / a, half) * MathS.Pow((2 * a - u) / a, half);
-            var inU = MathS.Pow(u, sumPower) * squared;
+            // u^n is A^n (u/A)^n up to a factor constant along the path, which the quotient below
+            // takes care of, and gathered with the secant's (u/A)^r one power: with a symbol for the
+            // powers, `u^(2 - n) (u/A)^(n - 1)` is (u/A)^1, and written apart it was read by no rule.
+            var gathered = (sumPower + half).InnerSimplified;
+            if (gathered is not Number && gathered.Complexity <= 40)
+                gathered = gathered.Simplify();
+            var inU = MathS.Pow(u / a, gathered) * MathS.Pow((2 * a - u) / a, half);
             if (Integration.ComputeAsAQuestionOfItsOwn(inU, u, integrateByParts) is not { } inUAnswer
                 || inUAnswer.Nodes.Any(node => node == MathS.NaN))
                 return null;

Sources/Tests/UnitTests/Calculus/SecantBesideAnImaginaryTangentSumIntegralTest.cs

@@ -31,13 +31,15 @@ public sealed class SecantBesideAnImaginaryTangentSumIntegralTest
         [InlineData("sec(x)^7/(a + i*a*tan(x))^4")]
         [InlineData("sec(x)^9/(a + i*a*tan(x))^8")]
         [InlineData("(k*sec(x))^3/(a - i*a*tan(x))^2")]
+        [InlineData("(m*sec(x))^(4 - 2*n)*(a + i*a*tan(x))^n")]
+        [InlineData("(m*sec(x))^(2*n)*(a + i*a*tan(x))^(3 - n)")]
         public void InTheSum(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("k", 0.7).Substitute("m", 1.1);
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("k", 0.7).Substitute("m", 1.1).Substitute("n", 2.3);
             var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
             var original = Pinned(integrand.ToEntity());
             var compared = 0;