An even power of the secant, cosine, sine or cosecant is written as the rational function of the tangent it is before the tangent substitution tests its rewriting

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

@@ -3494,6 +3494,28 @@ node is Cotanf(var cotangentArgument)
             if (!expr.ContainsNode(tangent))
                 return null;
 
+            // An even power of the secant, cosine, sine or cosecant of x is a rational function
+            // of the tangent, exactly and wherever it is defined: `sec^2 = 1 + tan^2`,
+            // `cos^2 = 1/(1 + tan^2)`, `sin^2 = tan^2/(1 + tan^2)`. Timofeev's
+            // `(sec^2 - 3 tan sqrt(4 sec^2 + 5 tan^2))/(sin^2 (4 sec^2 + 5 tan^2)^(3/2))` is
+            // `((1 + u^2) - 3u sqrt(4 + 9u^2))/(u^2 (4 + 9u^2)^(3/2))` under the tangent, and was
+            // declined for the secant and the sine that survived the rewriting. An odd power is
+            // not a function of the tangent -- its sign is not -- and is left, to survive and
+            // decline as before.
+            var secantSquared = 1 + MathS.Sqr(tangent);
+            expr = expr.Replace(node => node switch
+            {
+                Powf(Secantf(var a), Number.Integer even) when a == x && even.EInteger.IsEven && even.EInteger.Sign > 0
+                    => MathS.Pow(secantSquared, even.EInteger.ToInt32Checked() / 2),
+                Powf(Cosf(var a), Number.Integer even) when a == x && even.EInteger.IsEven && even.EInteger.Sign > 0
+                    => MathS.Pow(secantSquared, -(even.EInteger.ToInt32Checked() / 2)),
+                Powf(Sinf(var a), Number.Integer even) when a == x && even.EInteger.IsEven && even.EInteger.Sign > 0
+                    => MathS.Pow(MathS.Sqr(tangent) / secantSquared, even.EInteger.ToInt32Checked() / 2),
+                Powf(Cosecantf(var a), Number.Integer even) when a == x && even.EInteger.IsEven && even.EInteger.Sign > 0
+                    => MathS.Pow(secantSquared / MathS.Sqr(tangent), even.EInteger.ToInt32Checked() / 2),
+                _ => node,
+            });
+
             var uSub = Variable.CreateUnique(expr, "u_tan");
             var inU = expr.Substitute(tangent, uSub);
             if (inU.ContainsNode(x))

Sources/Tests/UnitTests/Calculus/ParityUnderASubstitutionTest.cs

@@ -97,15 +97,16 @@ public void TheOtherFunctionsAreWrittenAsSineAndCosine(string integrand)
         /// simplification the substitution runs takes <c>sqrt(.../cos(x)^2)</c> as
         /// <c>.../cos(x)</c>, its value where the cosine is positive and its negative
         /// elsewhere, and Timofeev's was answered wrongly on half the line that way. Declined
-        /// or right on both sides, never wrong.
+        /// or right on both sides, never wrong -- and answered now, by the tangent
+        /// substitution, which writes its even powers of the secant and the sine as the
+        /// rational functions of the tangent they are, exactly.
         /// </summary>
         [Theory]
         [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 AnOddComplementUnderARootIsNotWrittenSo(string integrand)
         {
             var integral = integrand.ToEntity().Integrate("x");
-            if (integral.Stringize().Contains("integral("))
-                return;   // declined is a legitimate verdict; a wrong answer is not
+            Assert.DoesNotContain("integral(", integral.Stringize());
             DifferentiatesBack(integrand, new[] { -2.6, -2.0, -0.4, 0.4, 2.0, 2.6 });
         }
 

Sources/Tests/UnitTests/Calculus/TangentSubstitutionIntegralTest.cs

@@ -82,6 +82,21 @@ private static void DifferentiatesBack(string integrand)
         [InlineData("tan(x)/(1 + tan(x)^2)")]
         public void AFunctionOfTheTangentAlone(string integrand) => DifferentiatesBack(integrand);
 
+        /// <summary>
+        /// An even power of the secant, cosine, sine or cosecant is a rational function of the
+        /// tangent, exactly and wherever it is defined, and is written as one before the test:
+        /// <c>sec^2 = 1 + tan^2</c>, <c>cos^2 = 1/(1 + tan^2)</c>, <c>sin^2 = tan^2/(1 + tan^2)</c>.
+        /// Timofeev's is <c>((1 + u^2) - 3u sqrt(4 + 9u^2))/(u^2 (4 + 9u^2)^(3/2))</c> under the
+        /// tangent, and was declined for the secant and the sine that survived. An odd power
+        /// is not a function of the tangent, and is left to decline as before.
+        /// </summary>
+        [Theory]
+        [InlineData("sec(x)^2*sqrt(tan(x))")]
+        [InlineData("sin(x)^2/(1 + tan(x))")]
+        [InlineData("csc(x)^4*tan(x)^(3/2)")]
+        [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>
         /// A power of the tangent, which the rewrite turns into an <b>improper</b> rational
         /// function — <c>u^2/(1 + u^2)</c> and <c>u^3/(1 + u^2)</c>. These were declined while