An odd power of the tangent beside a function of the secant, in the secant

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

@@ -4595,6 +4595,76 @@ static bool IsHalfOdd(Entity exponent) => exponent is Number.Rational rational a
             return expr * inTermsOfU.Substitute(u, other.Radicand) / inX;
         }
 
+        /// <summary>
+        /// An odd whole power of <c>tan(y)</c> beside a function of <c>sec(y)</c>, integrated in
+        /// <c>u = sec(y)</c>: <c>du = sec(y) tan(y) dy</c> and the even power of the tangent left over
+        /// is a power of <c>u^2 - 1</c>, so <c>tan(y)^n</c> with the rest is
+        /// <c>(u^2 - 1)^((n - 1)/2)/u du</c> beside it, exactly. The cotangent beside a function of
+        /// the cosecant, in <c>u = csc(y)</c>, the same.
+        /// </summary>
+        /// <remarks>
+        /// Rubi's <c>cot(c + d x)/(a + b sec(c + d x))^(3/2)</c> was declined and
+        /// <c>cot(c + d x)^3 sqrt(a + b sec(c + d x))</c> ran past thirty seconds, where in the
+        /// secant each is a rational function beside the root of a linear.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </remarks>
+        internal static Entity? SolveAnOddPowerOfTheTangentBesideAFunctionOfTheSecant(Entity expr, Entity.Variable x, bool integrateByParts)
+        {
+            if (!Integration.AnsweringTheQuestionAskedOrOneBelow)
+                return null;
+            Entity? argument = null;
+            foreach (var node in expr.Nodes)
+            {
+                if (TrigonometricArgument(node) is not { } thisArgument || !thisArgument.ContainsNode(x))
+                    continue;
+                if (argument is null)
+                    argument = thisArgument;
+                else if (argument != thisArgument)
+                    return null;
+            }
+            if (argument is null || !TreeAnalyzer.TryGetPolyLinear(argument, x, out var rate, out _)
+                || rate.ContainsNode(x) || TreeAnalyzer.IsZero(rate))
+                return null;
+            if (!expr.Nodes.All(node => !node.ContainsNode(x)
+                    || node is Variable or Sumf or Minusf or Mulf or Divf or Sinf or Cosf or Secantf or Cosecantf or Tanf or Cotanf
+                    || node is Powf(_, Number.Rational)))
+                return null;
+            var t = Variable.CreateUnique(expr, "t_odd_tangent");
+            var u = Variable.CreateUnique(expr, "u_secant");
+            foreach (var ofTheSecant in new[] { true, false })
+            {
+                // The tangent and its reciprocal as one base; for the cosecant, the cotangent.
+                var inT = expr.Replace(node => node switch
+                {
+                    Tanf(var a) when a == argument => ofTheSecant ? t : 1 / t,
+                    Cotanf(var a) when a == argument => ofTheSecant ? 1 / t : t,
+                    _ => node,
+                });
+                var (rest, exponents) = GatheredOverTheBases(inT, new Entity[] { t });
+                var n = exponents[0];
+                if (rest.ContainsNode(t) || !n.IsInteger() || n.ToEInteger().IsEven || !n.ToEInteger().CanFitInInt32())
+                    continue;
+                var inU = rest.Replace(node => node switch
+                {
+                    Secantf(var a) when ofTheSecant && a == argument => u,
+                    Cosf(var a) when ofTheSecant && a == argument => 1 / u,
+                    Cosecantf(var a) when !ofTheSecant && a == argument => u,
+                    Sinf(var a) when !ofTheSecant && a == argument => 1 / u,
+                    _ => node,
+                });
+                if (inU.ContainsNode(x))
+                    continue;
+                // dy = du/(u t) for the secant, -du/(u t) for the cosecant, and t^(n - 1) = (u^2 - 1)^((n - 1)/2).
+                var half = n.Subtract(ERational.One).Divide(ERational.FromInt32(2)).ToEInteger();
+                var integrand = (inU * MathS.Pow(MathS.Sqr(u) - 1, Number.Integer.Create(half)) / ((ofTheSecant ? rate : -rate) * u)).InnerSimplified;
+                if (Integration.ComputeAsAQuestionOfItsOwn(integrand, u, integrateByParts) is not { } inTermsOfU
+                    || inTermsOfU.Nodes.Any(node => node == MathS.NaN))
+                    return null;
+                return inTermsOfU.Substitute(u, ofTheSecant ? MathS.Sec(argument) : new Cosecantf(argument));
+            }
+            return null;
+        }
+
         /// <summary>
         /// <c>tan(y) tan(2y)</c> written <c>sec(2y) - 1</c>, and the integrand asked again in the
         /// one argument <c>2y</c>.

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

@@ -911,6 +911,8 @@ private static Entity Normalized(Entity expr, Entity.Variable x) =>
             // And the secant beside powers of a + a sec(y) and c - c sec(y), whose product is a
             // square of the tangent: in u = a + a sec(y), a symbolic power of it too.
             if ((answer = IndefiniteIntegralSolver.SolveASecantBesidePowersOfItsConjugateSums(expr, x, integrateByParts)) is { }) return answer;
+            // And an odd power of the tangent beside a function of the secant, in u = sec(y).
+            if ((answer = IndefiniteIntegralSolver.SolveAnOddPowerOfTheTangentBesideAFunctionOfTheSecant(expr, x, integrateByParts)) is { }) return answer;
             // And `a ± a cosh(y)` under a fractional power: `2a cosh(y/2)^2`, `-2a sinh(y/2)^2`.
             if ((answer = IndefiniteIntegralSolver.SolveByTheHalfAngleWhereOnePlusAHyperbolicCosineIsASquare(expr, x, integrateByParts)) is { }) return answer;
             // The hyperbolic sine's, off the real line: 1 + i sinh(y) is (cosh(y/2) + i sinh(y/2))^2.

Sources/Tests/UnitTests/Calculus/AnOddPowerOfTheTangentBesideTheSecantIntegralTest.cs

@@ -0,0 +1,50 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using System;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// An odd power of the tangent beside a function of the secant, in <c>u = sec(y)</c>, and the
+    /// cotangent beside one of the cosecant. Rubi's 4.5.1.4.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnOddPowerOfTheTangentBesideTheSecantIntegralTest
+    {
+        [Theory]
+        [InlineData("cot(g + f*x)/(a + b*sec(g + f*x))^(3/2)")]
+        [InlineData("cot(x)^3/(a + b*sec(x))^(3/2)")]
+        [InlineData("tan(x)^3/(a + b*csc(x))")]
+        public void InTheSecant(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 40000, $"{text.Length} characters of answer for {integrand}");
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.4).Substitute("g", 0.2).Substitute("f", 0.7);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -1.2, -0.7, 0.3, 0.8, 1.3, 2.9 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                if (want.IsNaN)
+                    continue;
+                compared++;
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart)
+                            < 1e-9 * Math.Max(1, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+            Assert.True(compared >= 5, $"only {compared} points could be compared for {integrand}");
+        }
+    }
+}