A symbolic power of a + i a tan beside a power of the secant is integrated as an exponential

BREAKING-CHANGES.md

@@ -945,6 +945,19 @@ wherever the cosine is negative; they are answered with the rest now
 | `"sqrt(c*sec(x))*sqrt(a + i*a*tan(x))".ToEntity().Integrate("x")` | `integral(...)` | the integrand times an antiderivative in `e^(i x/2)` over its derivative |
 | `"(c*sec(x))^(5/2)/(a + i*a*tan(x))^(5/2)".ToEntity().Integrate("x")` | `integral(...)` | the integrand times a power of `e^(i x)` |
 
+### A symbolic power of `a + i a tan` beside a power of the secant is integrated as an exponential
+
+**Answers where there were none.** `(k sec(c + d x))^(-4 - n) (a + i a tan(c + d x))^n` and the rest
+of Rubi's 4.3.1.2 whose two powers are symbols adding up to a whole number were declined: the powers
+were read as numbers only. In `w = e^(i z)` the integral is a power of `w` beside a whole power of
+`w^2 + 1`, a sum of powers of `w`
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(k*sec(c + d*x))^(-4 - n)*(a + i*a*tan(c + d*x))^n".ToEntity().Integrate("x")` | `integral(...)` | the integrand times powers of `e^(i (c + d x))` from `n - 4` to `n + 4`, each over its exponent |
+| `"(a + i*a*tan(c + d*x))^n/(k*sec(c + d*x))^n".ToEntity().Integrate("x")` | `integral(...)` | the integrand over `i d n` |
+
 ### A rational function of the tangent beside a power of `a + i a tan` is integrated in that sum
 
 **Answers where there were none.** `(A + B tan(c + d x))/sqrt(a + i a tan(c + d x))` and the rest of

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

@@ -26574,7 +26574,9 @@ bool ARootOfTheTangent(Entity node)
         /// cosine is positive, and was declined, with every other power of the pair whose exponents
         /// add up to a whole number: a root of a sum with the imaginary unit in it beside a root of the
         /// secant is read by no rule, and <c>e^(i x/2)/cos(x)</c>, what the positive sums leave, is
-        /// declined as well. Rubi's 4.3.1.2.
+        /// declined as well. Rubi's 4.3.1.2. A symbol for the powers is read too:
+        /// <c>(pe sec(z))^(-4 - n) (a + i a tan(z))^n</c> adds up to -4, and the integral in <c>w</c>
+        /// is <c>w^(n - 5) (w^2 + 1)^4</c>, a sum of powers.
         /// </para>
         /// <para>
         /// The constant is not written: the answer is the integrand times the antiderivative in
@@ -26591,7 +26593,7 @@ bool ARootOfTheTangent(Entity node)
                 return null;
             Entity? argument = null;
             var plus = false;
-            Number.Rational? tangentPower = null, secantPower = null;
+            Entity? tangentPower = null, secantPower = null;
             Entity constant = Number.Integer.One;
             Entity varying = Number.Integer.One;
             foreach (var (factor, underneath) in FactorsOfTheIntegrand(expr))
@@ -26602,11 +26604,13 @@ bool ARootOfTheTangent(Entity node)
                     continue;
                 }
                 varying = underneath ? varying / factor : varying * factor;
-                var (@base, power) = factor is Powf(var b, var p) && p.Evaled is Number.Rational r
-                    ? (b, r)
-                    : (factor, (Number.Rational)Number.Integer.One);
+                // A number or a symbol for the power: `(pe sec(z))^(-4 - n) (a + i a tan(z))^n`
+                // adds up to a whole number all the same.
+                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);
                 if (underneath)
-                    power = (Number.Rational)(-power);
+                    power = power is Number.Rational numeric ? -numeric : (-power).InnerSimplified;
                 if (TryReadAnImaginaryTangent(@base, x, out var tangentOf, out var isPlus))
                 {
                     if (tangentPower is not null || argument is not null && argument != tangentOf)
@@ -26627,22 +26631,34 @@ bool ARootOfTheTangent(Entity node)
             }
             if (tangentPower is null || secantPower is null || argument is null
                 || tangentPower is Number.Integer && secantPower is Number.Integer
-                || (tangentPower + secantPower) is not Number.Integer { EInteger: var whole } || !whole.CanFitInInt32()
+                || SumOfThePowers(tangentPower, secantPower) is not Number.Integer { EInteger: var whole } || !whole.CanFitInInt32()
                 || !TreeAnalyzer.TryGetPolyLinear(argument, x, out var slope, out _) || TreeAnalyzer.IsZero(slope))
                 return null;
             var k = whole.ToInt32Checked();
-            var phase = plus ? tangentPower : (Number.Rational)(-tangentPower);
+            var phase = plus ? tangentPower : (-tangentPower).InnerSimplified;
             // In w = e^(i z): sec(z)^k e^(i n z) dz is (2 w)^k (w^2 + 1)^(-k) w^(n - 1) dw / i.
             var w = Variable.CreateUnique(expr, "w_exp");
-            var inW = k >= 0
+            // At k = 0 the power of w alone: `(w^2 + 1)^0` is 1 only where `w^2 + 1` is not 0, and
+            // the integral of `w^(n - 1)` with that condition beside it was declined.
+            var inW = k == 0
+                ? MathS.Pow(w, (phase - 1).InnerSimplified)
+                : k > 0
                 ? MathS.Pow(2, k) * MathS.Pow(w, (phase + k - 1).InnerSimplified) / MathS.Pow(MathS.Sqr(w) + 1, k)
                 : MathS.Pow(2, k) * MathS.Pow(MathS.Sqr(w) + 1, -k) * MathS.Pow(w, (phase + k - 1).InnerSimplified);
             if (Integration.ComputeAsAQuestionOfItsOwn(inW.InnerSimplified, w, integrateByParts) is not { } inWAnswer
                 || inWAnswer.Nodes.Any(node => node == MathS.NaN))
                 return null;
+            // A short sum of symbols simplified, which the power rule leaves as it wrote it:
+            // `w^(n + -4 - 1 + 1)/(n + -4 - 1 + 1)` is `w^(n - 4)/(n - 4)`.
+            inWAnswer = inWAnswer.Replace(node => node is Sumf or Minusf && !node.ContainsNode(w) && node.Vars.Any() && node.Complexity <= 20
+                ? node.Simplify() : node);
             var exponential = MathS.Pow(MathS.e, MathS.i * argument);
             var differentiatesBackTo = MathS.Pow(MathS.Sec(argument), k) * MathS.Pow(exponential, phase);
             return constant * varying * inWAnswer.Substitute(w, exponential) / (MathS.i * slope * differentiatesBackTo);
+
+            // `n + (-4 - n)` is -4, which the inner simplification leaves as written.
+            static Entity SumOfThePowers(Entity first, Entity second)
+                => (first + second).InnerSimplified is var sum && (sum is Number || sum.Complexity > 40) ? sum : sum.Simplify();
         }
 
         /// <summary>

Sources/Tests/UnitTests/Calculus/ImaginaryTangentBesideTheSecantIntegralTest.cs

@@ -17,7 +17,8 @@ namespace AngouriMath.Tests.Calculus
     /// the real line, so the integrand is a constant on every interval where it is continuous times
     /// a power of the secant and an exponential. Rubi's 4.3.1.2. The integrands are complex along
     /// the real line and are compared as complex numbers, where the cosine is negative as well,
-    /// which is where the fourth row's answer on master was off by a constant factor.
+    /// which is where the fourth row's answer on master was off by a constant factor. A symbol
+    /// for the powers, adding up to a whole number, as well.
     /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
     /// </summary>
     [Trait("Area", "Calculus")]
@@ -29,12 +30,16 @@ public sealed class ImaginaryTangentBesideTheSecantIntegralTest
         [InlineData("(a + i*a*tan(x))^(3/2)/(k*sec(x))^(7/2)")]
         [InlineData("(k*sec(x))^(5/2)/(a + i*a*tan(x))^(5/2)")]
         [InlineData("(k*sec(c + d*x))^(3/2)*(a - i*a*tan(c + d*x))^(3/2)")]
+        // A symbol for the powers, adding up to -4, -1 and 0.
+        [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")]
         public void AsTheExponentialItIs(string integrand)
         {
             var integral = integrand.ToEntity().Integrate("x");
             Assert.DoesNotContain("integral(", integral.Stringize());
             Assert.DoesNotContain("NaN", integral.Stringize());
-            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("k", 0.8).Substitute("c", 0.4).Substitute("d", 1.1);
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("k", 0.8).Substitute("c", 0.4).Substitute("d", 1.1).Substitute("n", 0.6);
             var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
             var original = Pinned(integrand.ToEntity());
             var compared = 0;