A power of the cosine beside a power of a + i a tan is read as one of the secant

BREAKING-CHANGES.md

@@ -382,6 +382,20 @@ right answers were declined after seconds
 | `"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 |
 
+### A power of the cosine beside a power of `a + i a tan` is read as one of the secant
+
+**Answers where there were none, and shorter ones.** `sqrt(a + i a tan(x))/(k cos(x))^(3/2)` was declined:
+the rule that integrates a power of `a + i a tan(z)` beside a power of the secant as the exponential it is,
+where the powers add up to a whole number, read the secant only, and the cosine below the bar is the same
+power of it with its sign turned. It reads the cosine too now, the factors kept as they are written; the
+rows the rule for any powers answered in `u` before are answered by it instead, and shorter
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sqrt(a + i*a*tan(x))/(k*cos(x))^(3/2)".ToEntity().Integrate("x")` | `integral(...)` | in `e^(i x)`, 533 characters |
+| `"(k*cos(x))^(3/2)*sqrt(a + i*a*tan(x))".ToEntity().Integrate("x")` | `integral(...)`; 763 characters on the unreleased master | 172 characters |
+
 ### A power of the secant beside a power of `a + i a tan` is integrated in the sum, whatever the powers
 
 **Answers where there were none.** `sec(x)^3 sqrt(a + i a tan(x))` was declined, with the rest of

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

@@ -26935,7 +26935,7 @@ bool ARootOfTheTangent(Entity node)
         /// </remarks>
         internal static Entity? SolveAPowerOfAnImaginaryTangentBesideAPowerOfTheSecant(Entity expr, Entity.Variable x, bool integrateByParts)
         {
-            if (!expr.Nodes.Any(node => node is Secantf) || !expr.Nodes.Any(node => node is Tanf))
+            if (!expr.Nodes.Any(node => node is Secantf or Cosf) || !expr.Nodes.Any(node => node is Tanf))
                 return null;
             Entity? argument = null;
             var plus = false;
@@ -26964,16 +26964,21 @@ bool ARootOfTheTangent(Entity node)
                     (tangentPower, argument, plus) = (power, tangentOf, isPlus);
                     continue;
                 }
-                var secant = @base switch
+                // A power of the cosine is one of the secant with the sign of the power turned:
+                // `sqrt(a + i a tan(z))/(e cos(z))^(3/2)` adds up to 2 as `sqrt(a + i a tan(z)) (e sec(z))^(3/2)`
+                // does, the factors themselves kept as they are written.
+                var (trigonometric, turned) = @base switch
                 {
-                    Secantf => @base,
-                    Mulf(var left, Secantf right) when !left.ContainsNode(x) => right,
-                    Mulf(Secantf left, var right) when !right.ContainsNode(x) => left,
-                    _ => null
+                    Secantf => (@base, false),
+                    Cosf => (@base, true),
+                    Mulf(var left, var right) when !left.ContainsNode(x) && right is Secantf or Cosf => (right, right is Cosf),
+                    Mulf(var left, var right) when !right.ContainsNode(x) && left is Secantf or Cosf => (left, left is Cosf),
+                    _ => ((Entity?)null, false)
                 };
-                if (secant is not Secantf(var secantOf) || secantPower is not null || argument is not null && argument != secantOf)
+                var secantOf = trigonometric switch { Secantf(var inner) => inner, Cosf(var inner) => inner, _ => null };
+                if (secantOf is null || secantPower is not null || argument is not null && argument != secantOf)
                     return null;
-                (secantPower, argument) = (power, secantOf);
+                (secantPower, argument) = (turned ? (power is Number.Rational turnedPower ? -turnedPower : (-power).InnerSimplified) : power, secantOf);
             }
             if (tangentPower is null || secantPower is null || argument is null
                 || tangentPower is Number.Integer && secantPower is Number.Integer

Sources/Tests/UnitTests/Calculus/ImaginaryTangentBesideTheSecantIntegralTest.cs

@@ -34,6 +34,8 @@ public sealed class ImaginaryTangentBesideTheSecantIntegralTest
         [InlineData("(k*sec(c + d*x))^(-4 - n)*(a + i*a*tan(c + d*x))^n")]
         [InlineData("(k*sec(c + d*x))^(-1 - n)*(a + i*a*tan(c + d*x))^n")]
         [InlineData("(a + i*a*tan(c + d*x))^n/(k*sec(c + d*x))^n")]
+        [InlineData("sqrt(a + i*a*tan(x))/(k*cos(x))^(3/2)")]
+        [InlineData("(k*cos(c + d*x))^(5/2)/sqrt(a + i*a*tan(c + d*x))")]
         public void AsTheExponentialItIs(string integrand)
         {
             var integral = integrand.ToEntity().Integrate("x");