An even power of the secant or the cosecant under a root is written in the tangent (#1717)

The tangent substitution writes sec^2 as 1 + tan^2 and csc^2 as
(1 + tan^2)/tan^2, but went on only where the tangent, or a sine or a
cosine under a root, was already in the integrand, so sqrt(a + b csc(x)^2)
was declined where sqrt(a + b cot(x)^2) was answered. An even power of
the secant or the cosecant under a root goes on now too. Rubi's 4.5.7
and 4.6.7.

Part of #718


Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

Co-authored-by: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

Author: Rafael-SOWNet

BREAKING-CHANGES.md

@@ -896,6 +896,26 @@ its sign on its own, so an answer through two says `provided cos(y) >= 0`
 | `"sqrt(a - a*sec(x))".ToEntity().Integrate("x")` | `integral(...)` | a logarithm in `tan(x/2)`, times `sgn(tan(x/2))` |
 | `"sqrt(1 + csc(x))".ToEntity().Integrate("x")` | `integral(...)` | an arctangent in `tan(pi/4 - x/2)` |
 
+### An even power of the secant or the cosecant under a root is written in the tangent
+
+**Answers where there were none.** `sqrt(a + b csc(x)^2)`, `(a + b sec(x)^2)^(3/2)` and their kin,
+Rubi's 4.5.7 and 4.6.7, were declined, while `sqrt(a + b tan(x)^2)` and `sqrt(a + b cot(x)^2)` were
+answered through `u = tan(x)`. That substitution writes an even power of the secant or the
+cosecant in the tangent, `sec^2 = 1 + tan^2` and `csc^2 = (1 + tan^2)/tan^2`, but went on to do so
+only where the tangent, or a sine or a cosine under a root, was already in the integrand. An even
+power of the secant or the cosecant under a root goes on now too. The cosecant's answers carry
+`sgn(tan(x))`, from the root of `1/tan(x)^2`
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sqrt(a + b*csc(x)^2)".ToEntity().Integrate("x")` | `integral(...)` | arctangents in `sqrt((a + b) tan(x)^2 + b)`, times `sgn(tan(x))` |
+| `"(a + b*csc(c + d*x)^2)^(3/2)".ToEntity().Integrate("x")` | `integral(...)` | the same in `tan(c + d x)` and an algebraic part, by the signs of `a` and `b` |
+| `"sqrt(a + b*sec(x)^2)".ToEntity().Integrate("x")` | `integral(...)` | arctangents in `tan(x)/sqrt(a + b (1 + tan(x)^2))` |
+| `"1/sqrt(a + b*sec(x)^2)".ToEntity().Integrate("x")` | `integral(...)` | an arctangent or a logarithm in the same, by the sign of `a` |
+| `"sqrt(1 + csc(x)^2)".ToEntity().Integrate("x")` | `integral(...)` | logarithms and an arctangent in `sqrt(2 tan(x)^2 + 1)`, times `sgn(tan(x))` |
+| `"1/sqrt(-1 + csc(x)^2)".ToEntity().Integrate("x")` | `integral(...)` | `sgn(tan(x)) ln(1 + tan(x)^2)/2` |
+
 ### A partial-fraction coefficient with symbols in it is in lowest terms, its rational content included
 
 **Improvement, not silent.** The decomposition over written factors with symbols among their

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

@@ -8464,9 +8464,13 @@ node is Powf(var product, Number.Rational fraction) && fraction is not Number.In
                     : node);
 
             var tangent = MathS.Tan(x);
-            // Or a sine or a cosine under a root, for the writing by the sign below; a rational
-            // function of those is the half-angle substitution's.
-            if (!expr.ContainsNode(tangent) && !(HasARadicalOf(expr, x) && expr.Nodes.Any(node => node is Sinf or Cosf && node.ContainsNode(x))))
+            // Or a sine or a cosine under a root, for the writing by the sign below, or an even
+            // power of the secant or the cosecant under one, which the next step writes in the
+            // tangent: `sqrt(a + b csc(x)^2)` is `sqrt(a + b + b/tan(x)^2)`. A rational function
+            // of those is the half-angle substitution's.
+            if (!expr.ContainsNode(tangent) && !(HasARadicalOf(expr, x) && expr.Nodes.Any(node =>
+                    node is Sinf or Cosf && node.ContainsNode(x)
+                    || node is Powf(Secantf or Cosecantf, Number.Integer { EInteger.IsEven: true }) && node.ContainsNode(x))))
                 return null;
 
             // An even power of the secant, cosine, sine or cosecant of x is a rational function

Sources/Tests/UnitTests/Calculus/TangentSubstitutionIntegralTest.cs

@@ -97,6 +97,51 @@ private static void DifferentiatesBack(string integrand)
         [InlineData("(sec(x)^2 - 3*sqrt(4*sec(x)^2 + 5*tan(x)^2)*tan(x))/(sin(x)^2*(4*sec(x)^2 + 5*tan(x)^2)^(3/2))")]
         public void AnEvenPowerOfTheOthersIsWrittenInTheTangent(string integrand) => DifferentiatesBack(integrand);
 
+        /// <summary>
+        /// The same under a root with no tangent beside it, Rubi's 4.5.7 and 4.6.7:
+        /// <c>sqrt(a + b csc(x)^2)</c> is <c>sqrt(a + b (1 + u^2)/u^2)</c> under the tangent. On both
+        /// signs of the tangent, which the cosecant's answer reads, with the symbols pinned before
+        /// the derivative is taken: <c>sgn(tan(c + d x))</c> is flat only where <c>c</c> and
+        /// <c>d</c> are real.
+        /// </summary>
+        [Theory]
+        [InlineData("sqrt(a + b*csc(x)^2)")]
+        [InlineData("(a + b*csc(x)^2)^(3/2)")]
+        [InlineData("1/(a + b*csc(x)^2)^(3/2)")]
+        [InlineData("sqrt(a + b*sec(x)^2)")]
+        [InlineData("1/sqrt(a + b*sec(x)^2)")]
+        [InlineData("sqrt(a + b*csc(c + d*x)^2)")]
+        [InlineData("sqrt(1 + csc(x)^2)")]
+        [InlineData("1/sqrt(-1 + csc(x)^2)")]
+        public void AnEvenPowerOfTheSecantOrTheCosecantUnderARoot(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            var pins = new (string, double)[] { ("a", 1.3), ("b", 0.7), ("c", 0.4), ("d", 1.1) };
+            var answer = integral.Substitute("C", 0);
+            var original = integrand.ToEntity();
+            foreach (var (name, value) in pins)
+            {
+                answer = answer.Substitute(name, value);
+                original = original.Substitute(name, value);
+            }
+            var derivative = answer.Differentiate("x");
+            var compared = 0;
+            foreach (var at in new[] { -2.6, -0.9, 0.4, 1.1, 1.9, 2.7 })
+            {
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                var want = original.Substitute("x", at).EvalNumerical();
+                if (got.IsNaN || want.IsNaN)
+                    continue;
+                compared++;
+                var difference = Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart);
+                var scale = Math.Max(1.0, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart));
+                Assert.True(difference / scale < 1e-9,
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+            Assert.True(compared >= 4, $"only {compared} points could be compared for {integrand}");
+        }
+
         /// <summary>
         /// An odd power of the sine or the cosine is its sign times a function of the tangent,
         /// <c>cos(x) = sgn(cos(x))/sqrt(1 + tan^2)</c> and <c>sin(x) = tan(x) cos(x)</c>, the sign a