Two tangents of arguments a constant apart are written apart

BREAKING-CHANGES.md

@@ -826,6 +826,19 @@ read them as nonzero. They are now decided over one bar, expanded, and at pinned
 | `"e^(3*acoth(a*x))/(c-a*c*x)^3".Integrate("x")`, and over `(c - a c x)^4` | left unevaluated | an antiderivative in `sqrt((a x + 1)/(a x - 1))` |
 | `"e^(2*acoth(a*x))*sqrt(c-a*c*x)/x".Integrate("x")`, and over `x^2` | left unevaluated | an antiderivative in `sqrt(c - a c x)`, by cases on the sign of `c` |
 
+### Two tangents of arguments a constant apart are written apart
+
+**Answers where there were none.** `tan(a + b x) tan(c + b x)` was declined, with the secants,
+cotangents and cosecants the same way and the products whose arguments add to a constant, Rubi's
+4.7.7. By the addition formulas each such product is the functions of the two arguments apart:
+`tan(A) tan(B) = cot(A - B) (tan(A) - tan(B)) - 1` and its kin, for `A - B` or `A + B` a constant
+that is not a multiple of `pi` ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"tan(a + b*x)*tan(c + b*x)".ToEntity().Integrate("x")` | `integral(...)` | `cot(a - c) (ln(cos(c + b x)) - ln(cos(a + b x)))/b - x` |
+| `"sec(c - b*x)*sec(a + b*x)".ToEntity().Integrate("x")` | `integral(...)` | `csc(a + c) (ln(cos(c - b x)) - ln(cos(a + b x)))/b` |
+
 ### A trigonometric function of an imaginary multiple of a logarithm is integrated in exponentials
 
 **Answers where there were none.** `tan(a + i ln(x))` and `sin(a + ln(c x^2) sqrt(-1/4))` were

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

@@ -8809,6 +8809,76 @@ private static (Entity Coefficient, Entity Degree)? TheMonomial(Entity expr, Ent
             return null;
         }
 
+        /// <summary>
+        /// A product of two tangents, cotangents, secants or cosecants of linear arguments whose
+        /// difference or sum is a constant, written as the functions of each apart: with
+        /// <c>d = A - B</c>, <c>tan(A) tan(B) = cot(d) (tan(A) - tan(B)) - 1</c>.
+        /// </summary>
+        /// <remarks>
+        /// <para>
+        /// The addition formulas, read for the product: for a constant <c>d = A - B</c>,
+        /// <c>tan(A) tan(B) = cot(d) (tan(A) - tan(B)) - 1</c>,
+        /// <c>cot(A) cot(B) = cot(d) (cot(B) - cot(A)) - 1</c>,
+        /// <c>sec(A) sec(B) = csc(d) (tan(A) - tan(B))</c> and
+        /// <c>csc(A) csc(B) = csc(d) (cot(B) - cot(A))</c>; for a constant <c>s = A + B</c>,
+        /// <c>tan(A) tan(B) = 1 - cot(s) (tan(A) + tan(B))</c>,
+        /// <c>cot(A) cot(B) = 1 + cot(s) (cot(A) + cot(B))</c>,
+        /// <c>sec(A) sec(B) = csc(s) (tan(A) + tan(B))</c> and
+        /// <c>csc(A) csc(B) = csc(s) (cot(A) + cot(B))</c>. Each holds wherever both sides are
+        /// defined, for <c>d</c> or <c>s</c> not a multiple of <c>pi</c>, which a symbolic one is
+        /// taken not to be, as everywhere in this integrator. Rubi's 4.7.7 has
+        /// <c>tan(a + b x) tan(c + b x)</c> and the rest, and nothing read two arguments.
+        /// </para>
+        /// <para>
+        /// The same question in another spelling, so asked as it. A constant times the product
+        /// only, and the two functions of one kind.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </para>
+        /// </remarks>
+        internal static Entity? SolveByWritingTwoFunctionsOfShiftedArgumentsApart(Entity expr, Entity.Variable x, bool integrateByParts)
+        {
+            Entity constant = Number.Integer.One;
+            Entity? first = null, second = null;
+            foreach (var factor in Mulf.LinearChildren(expr))
+            {
+                if (!factor.ContainsNode(x))
+                    constant = constant == Number.Integer.One ? factor : constant * factor;
+                else if (first is null)
+                    first = factor;
+                else if (second is null)
+                    second = factor;
+                else
+                    return null;
+            }
+            if (first is null || second is null || first.GetType() != second.GetType()
+                || first is not (Tanf or Cotanf or Secantf or Cosecantf))
+                return null;
+            var a = first.DirectChildren.First();
+            var b = second.DirectChildren.First();
+            if (a == b || !TreeAnalyzer.TryGetPolyLinear(a, x, out var slopeOfA, out _) || slopeOfA.ContainsNode(x)
+                || !TreeAnalyzer.TryGetPolyLinear(b, x, out var slopeOfB, out _) || slopeOfB.ContainsNode(x))
+                return null;
+            static bool IsZero(Entity value) => value.Expand().InnerSimplified.Evaled is Number.Complex { IsZero: true };
+            var sameSlope = IsZero(slopeOfA - slopeOfB);
+            if (!sameSlope && !IsZero(slopeOfA + slopeOfB))
+                return null;
+            var shift = (sameSlope ? a - b : a + b).Expand().InnerSimplified;
+            if (shift.ContainsNode(x) || shift.Evaled is Number.Complex { IsZero: true })
+                return null;
+            Entity apart = first switch
+            {
+                Tanf => sameSlope
+                    ? MathS.Cotan(shift) * (MathS.Tan(a) - MathS.Tan(b)) - 1
+                    : 1 - MathS.Cotan(shift) * (MathS.Tan(a) + MathS.Tan(b)),
+                Cotanf => sameSlope
+                    ? MathS.Cotan(shift) * (MathS.Cotan(b) - MathS.Cotan(a)) - 1
+                    : 1 + MathS.Cotan(shift) * (MathS.Cotan(a) + MathS.Cotan(b)),
+                Secantf => MathS.Cosec(shift) * (sameSlope ? MathS.Tan(a) - MathS.Tan(b) : MathS.Tan(a) + MathS.Tan(b)),
+                _ => MathS.Cosec(shift) * (sameSlope ? MathS.Cotan(b) - MathS.Cotan(a) : MathS.Cotan(a) + MathS.Cotan(b)),
+            };
+            return Integration.ComputeAsTheSameQuestion(constant == Number.Integer.One ? apart : constant * apart, x, integrateByParts);
+        }
+
         /// <summary>
         /// A quotient of two <b>homogeneous</b> polynomials in <c>sin(u)</c> and <c>cos(u)</c>,
         /// integrated by <c>t = tan(u)</c> — which turns it into a rational function of <c>t</c>

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

@@ -884,6 +884,9 @@ private static Entity Normalized(Entity expr, Entity.Variable x) =>
             // search, which reads the quotient term by term.
             if ((answer = IndefiniteIntegralSolver.SolveACosineAndASineOverAPowerOfAnother(expr, x, integrateByParts)) is { }) return answer;
             if ((answer = IndefiniteIntegralSolver.SolveByWritingMultiplesOfOneLinearArgument(expr, x, integrateByParts)) is { }) return answer;
+            // Two tangents, cotangents, secants or cosecants of arguments a constant apart, written
+            // as functions of each alone by the addition formulas.
+            if ((answer = IndefiniteIntegralSolver.SolveByWritingTwoFunctionsOfShiftedArgumentsApart(expr, x, integrateByParts)) is { }) return answer;
             // A constant out of a fractional power of a trigonometric factor: `sqrt(b sec(x))` is
             // `sqrt(b) sqrt(sec(x))` for a positive `b`, which meets the other powers of the
             // secant beside it. After the rules that answer the same shapes for any real

Sources/Tests/UnitTests/Calculus/TwoFunctionsOfShiftedArgumentsIntegralTest.cs

@@ -0,0 +1,52 @@
+//
+// 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>
+    /// Two tangents, cotangents, secants or cosecants of arguments whose difference or sum is a
+    /// constant, written as the functions of each apart by the addition formulas. Rubi's 4.7.7.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    /// <remarks>
+    /// Checked by differentiating back with <c>a = 0.3</c>, <c>b = 0.9</c>, <c>c = 1.1</c>, on
+    /// both sides of zero and away from the poles of every function of the three arguments.
+    /// </remarks>
+    [Trait("Area", "Calculus")]
+    public sealed class TwoFunctionsOfShiftedArgumentsIntegralTest
+    {
+        [Theory]
+        [InlineData("tan(a + b*x)*tan(c + b*x)")]
+        [InlineData("tan(c - b*x)*tan(a + b*x)")]
+        [InlineData("cot(a + b*x)*cot(c + b*x)")]
+        [InlineData("cot(c - b*x)*cot(a + b*x)")]
+        [InlineData("sec(a + b*x)*sec(c + b*x)")]
+        [InlineData("sec(c - b*x)*sec(a + b*x)")]
+        [InlineData("csc(a + b*x)*csc(c + b*x)")]
+        [InlineData("csc(c - b*x)*csc(a + b*x)")]
+        public void IsWrittenApartByTheAdditionFormulas(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Entity Pinned(Entity e) => e.Substitute("a", 0.3).Substitute("b", 0.9).Substitute("c", 1.1);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            foreach (var at in new[] { -1.0, -0.8, 0.1, 0.2, 0.9, 1.0 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                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}");
+            }
+        }
+    }
+}