a + i a tan below an odd power of the secant is written over its conjugate

BREAKING-CHANGES.md

@@ -365,6 +365,21 @@ quotient of two such linears the sum is `(b - d t^2)^2 + (c t^2 - a)^2`. Rubi's
 | `"(c + d*tan(x))^(3/2)/(a + b*tan(x))^3".ToEntity().Integrate("x")` | `integral(...)` | the same |
 | `"1/((a + b*tan(x))^(3/2)*(c + d*tan(x))^(3/2))".ToEntity().Integrate("x")` | `integral(...)`; past a minute on the unreleased master | in the root of the quotient of the two, in a second |
 
+### `a + i a tan` below an odd power of the secant is written over its conjugate
+
+**Answers where there were none, and shorter ones.** `sec(x)^7/(a + i a tan(x))^4` was declined, with
+the rest of Rubi's 4.3.1.2 that is an odd power of the secant over a whole power of `a + i a tan`. The sum
+times `a - i a tan(z)` is `a^2 sec(z)^2`, so `sec(z)^s/(a + i a tan(z))^n` is
+`sec(z)^(s - 2 n) (a - i a tan(z))^n/a^(2 n)`, a whole power of the conjugate beside one of the secant,
+and it is integrated as that. Those integrated in `u = a + i a tan(z)` before come out a tenth as long
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sec(x)^7/(a + i*a*tan(x))^4".ToEntity().Integrate("x")` | `integral(...)` | 371 characters |
+| `"sec(x)^9/(a + i*a*tan(x))^8".ToEntity().Integrate("x")` | `integral(...)` | 745 characters |
+| `"sec(x)^5/(a + i*a*tan(x))^2".ToEntity().Integrate("x")` | `integral(...)`; 2,431 characters on the unreleased master | 225 characters |
+
 ### An odd power of the secant over a whole power of `a + i a tan` is integrated in the sum
 
 **Answers where there were none.** `sec(x)^5/(a + i a tan(x))^2` was declined: the rule that integrates
@@ -379,8 +394,8 @@ right answers were declined after seconds
 
 | Input | Was (2.5.0) | Now |
 |---|---|---|
-| `"sec(x)^5/(a + i*a*tan(x))^2".ToEntity().Integrate("x")` | `integral(...)` | 2,431 characters |
-| `"sec(x)^7/(a + i*a*tan(x))^3".ToEntity().Integrate("x")` | `integral(...)` | 3,774 characters |
+| `"sec(x)^5/(a + i*a*tan(x))^2".ToEntity().Integrate("x")` | `integral(...)` | 225 characters |
+| `"sec(x)^7/(a + i*a*tan(x))^3".ToEntity().Integrate("x")` | `integral(...)` | 289 characters |
 
 ### A power of the cosine beside a power of `a + i a tan` is read as one of the secant
 

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

@@ -27047,10 +27047,15 @@ static Entity SumOfThePowers(Entity first, Entity second)
             Entity? sumPower = null;
             Entity secantPower = Number.Integer.Zero;
             var sawSecant = false;
+            // Everything but the sum, for the rewrite of a whole power of it below.
+            Entity others = Number.Integer.One;
             foreach (var (factor, underneath) in FactorsOfTheIntegrand(expr))
             {
                 if (!factor.ContainsNode(x))
+                {
+                    others = underneath ? others / factor : others * factor;
                     continue;
+                }
                 var (@base, power) = factor is Powf(var b, var p) && !p.ContainsNode(x)
                     ? (b, p.Evaled is Number.Rational r ? r : p)
                     : (factor, (Entity)Number.Integer.One);
@@ -27077,27 +27082,41 @@ static Entity SumOfThePowers(Entity first, Entity second)
                 if (of is null || argument is not null && argument != of)
                     return null;
                 argument = of;
+                others = underneath ? others / factor : others * factor;
                 secantPower = sign == 1 ? secantPower + power : secantPower - power;
                 sawSecant = true;
             }
             if (sumPower is null || sum is null || argument is null || !sawSecant
                 || !TreeAnalyzer.TryGetPolyLinear(argument, x, out var slope, out _) || TreeAnalyzer.IsZero(slope))
                 return null;
             secantPower = secantPower is Number ? secantPower : secantPower.InnerSimplified;
-            // Whole powers on both are left to the rules for the sine and the cosine, which answer
-            // `cos(x)^5/(a + i a tan(x))^3` and `sec(x)^3/(a + i a tan(x))^4` in a fraction of a second
-            // where in u they ran past five -- but for an odd power s of the secant over the sum's n-th
-            // with s + 2 n = 1. Then u^n (sec(z)^2)^((s - 2)/2) is a whole power of 2 A - u over the root
-            // of u, which is answered in a second, and `sec(x)^5/(a + i a tan(x))^2` was declined by
-            // every route but this.
-            if (sumPower is Number.Integer { EInteger: var whole } && secantPower is Number.Integer { EInteger: var secant }
-                && !(secant + whole * 2).Equals(EInteger.One))
-                return null;
             Entity constantTerm = Number.Integer.Zero;
             foreach (var term in Sumf.LinearChildren(sum))
                 if (!term.ContainsNode(x))
                     constantTerm += term;
             var a = constantTerm.InnerSimplified;
+            // Whole powers on both. An odd power s of the secant over the sum's n-th: the sum times its
+            // conjugate is A^2 sec(z)^2, so `sec(x)^7/(a + i a tan(x))^4` is
+            // `sec(x)^(-1) (a - i a tan(x))^4/a^8`, a whole power of the conjugate beside one of the
+            // secant, which the rules for the sine and the cosine answer in a fraction of a second. In u
+            // it ran past five, or with s + 2 n = 1 took two seconds for an answer ten times as long.
+            // An even power is left to those rules as written, which answer `sec(x)^4/(a + i a tan(x))^3`
+            // shorter than its rewrite, and so is an odd power of the cosine, `cos(x)/(a + i a tan(x))^4`,
+            // which they answer several times sooner; and a whole power above the bar to them too, but for
+            // s + 2 n = 1, where u^n (sec(z)^2)^((s - 2)/2) is a whole power of 2 A - u over the root of u.
+            if (sumPower is Number.Integer { EInteger: var whole } && secantPower is Number.Integer { EInteger: var secant })
+            {
+                if (whole.Sign < 0 && secant.Sign > 0 && !secant.IsEven && whole.CanFitInInt32())
+                {
+                    var n = -whole.ToInt32Checked();
+                    var conjugate = a + (plus ? -MathS.i : MathS.i) * a * MathS.Tan(argument);
+                    return Integration.ComputeAsTheSameQuestion(
+                        (others * MathS.Pow(conjugate, n) * MathS.Pow(MathS.Sec(argument), -2 * n) / MathS.Pow(a, 2 * n)).InnerSimplified,
+                        x, integrateByParts);
+                }
+                if (!(secant + whole * 2).Equals(EInteger.One))
+                    return null;
+            }
             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);

Sources/Tests/UnitTests/Calculus/SecantBesideAnImaginaryTangentSumIntegralTest.cs

@@ -28,6 +28,9 @@ public sealed class SecantBesideAnImaginaryTangentSumIntegralTest
         [InlineData("cos(x)^9*(a + i*a*tan(x))^(7/2)")]
         [InlineData("(m*sec(x))^(2/3)*(a + i*a*tan(x))^(5/3)")]
         [InlineData("sec(x)^5/(a + i*a*tan(x))^2")]
+        [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")]
         public void InTheSum(string integrand)
         {
             var integral = integrand.ToEntity().Integrate("x");