An odd power of the secant is read in the tangent, with the cosine's sign

BREAKING-CHANGES.md

@@ -270,6 +270,20 @@ answer on the far side of a root gets a real expression where it used to get a c
 | `1/(x*(-4+x^2)^4)` | 8,472 ms | 1,878 ms |
 | `1/((1+x)^3*(2+x)^3)` | 1,343 ms | 492 ms |
 
+### An odd power of the secant is read in the tangent, with the cosine's sign
+
+**Answers where there were none.** `sec(x)^5/(a + b tan(x))^2` was declined: the substitution
+`u = tan(x)` read an odd power of the sine or the cosine as its sign times a function of the tangent,
+`cos(x) = sgn(cos(x))/sqrt(1 + u^2)`, and not the secant, which is the same the other way up. It reads the
+secant too now, and the integrand is `sgn(cos(x)) (1 + u^2)^(3/2)/(a + b u)^2` in `u`
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sec(x)^5/(a + b*tan(x))^2".ToEntity().Integrate("x")` | `integral(...)` | in `sgn(cos(x))` and `tan(x)`, 901 characters |
+| `"sec(x)^7/(a + b*tan(x))^3".ToEntity().Integrate("x")` | `integral(...)` | 1,253 characters |
+| `"sec(x)/(a + b*cot(x))".ToEntity().Integrate("x")` | `integral(...)` | 571 characters |
+
 ### A whole power of the radicand under Euler's substitution is read as one
 
 **Answers where there were none, and none where they were wrong.** Euler's substitution writes the

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

@@ -10283,6 +10283,11 @@ static bool IsTwice(Entity argument, Entity.Variable x)
                         => MathS.Pow(sign / rootOfSecantSquared, k),
                     Powf(Sinf(var a), Number.Integer k) when a == x && k.EInteger.CanFitInInt32() && !k.EInteger.IsEven
                         => MathS.Pow(sign * tangent / rootOfSecantSquared, k),
+                    // And the secant is the cosine the other way up, the sign its own inverse:
+                    // `sec(x)^5/(a + b tan(x))^2` is `sgn(cos(x)) (1 + u^2)^(5/2)/(a + b u)^2`.
+                    Secantf(var a) when a == x => sign * rootOfSecantSquared,
+                    Powf(Secantf(var a), Number.Integer k) when a == x && k.EInteger.CanFitInInt32() && !k.EInteger.IsEven
+                        => MathS.Pow(sign * rootOfSecantSquared, k),
                     _ => node,
                 }).Substitute(tangent, uSub);
                 if (odd.ContainsNode(x) || !IsAlgebraicIn(odd, uSub))

Sources/Tests/UnitTests/Calculus/TangentSubstitutionIntegralTest.cs

@@ -113,7 +113,20 @@ private static void DifferentiatesBack(string integrand)
         [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)
+        public void AnEvenPowerOfTheSecantOrTheCosecantUnderARoot(string integrand) => DifferentiatesBackWithTheSymbolsPinned(integrand);
+
+        /// <summary>
+        /// An odd power of the secant is the cosine's the other way up: <c>sec(x)^5/(a + b tan(x))^2</c>
+        /// is <c>sgn(cos(x)) (1 + u^2)^(3/2)/(a + b u)^2</c> under <c>u = tan(x)</c>. Rubi's 4.3.1.2,
+        /// checked on both signs of the cosine.
+        /// </summary>
+        [Theory]
+        [InlineData("sec(x)^5/(a + b*tan(x))^2")]
+        [InlineData("sec(x)^7/(a + b*tan(x))^3")]
+        [InlineData("sec(c + d*x)^5/(a + b*tan(c + d*x))^3")]
+        public void AnOddPowerOfTheSecantIsItsSignTimesAFunctionOfTheTangent(string integrand) => DifferentiatesBackWithTheSymbolsPinned(integrand);
+
+        private static void DifferentiatesBackWithTheSymbolsPinned(string integrand)
         {
             var integral = integrand.ToEntity().Integrate("x");
             Assert.DoesNotContain("integral(", integral.Stringize());