Read the symbolic trigonometric power as a factor of the integrand

BREAKING-CHANGES.md

@@ -910,13 +910,20 @@ in Rubi's 1.1.3.4 come out shorter as well, and those that took two seconds take
 of Rubi's 4.1.4.1, 4.2.4.1 and 4.5.4.1. `f^m (alpha + beta f^2)` for `f` a sine, cosine, secant or cosecant of a linear
 is the derivative of `cos sin^(m + 1)`, `sin cos^(m + 1)`, `tan sec^m` or `cot csc^m` where its bracket is a multiple of
 that derivative's, and is now integrated so ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+The power is read as a factor of the integrand, so one under the bar is `f^(-m)`: `(m + (1 - m) sec^2)/sec^m` is
+the derivative of `tan sec^(-m)`. And `m` may be a number that is not whole, as in Rubi's 4.5.4.1
+`(3 + 3 sec(c + d x)^2)/sqrt(sec(c + d x))`.
 
 | Input | Was (2.5.0) | Now |
 |---|---|---|
 | `"sec(c+d*x)^m*(-q*m/(1+m)+q*sec(c+d*x)^2)".ToEntity().Integrate("x")` | `integral(...)` | 75 characters |
 | `"sec(c+d*x)^m*(A-A*(1+m)*sec(c+d*x)^2/m)".ToEntity().Integrate("x")` | `integral(...)` | 54 characters |
 | `"sec(k+f*x)^m*(m-(1+m)*sec(k+f*x)^2)".ToEntity().Integrate("x")` | `integral(...)` | 54 characters |
 | `"sin(k+f*x)^m*(1+m-(2+m)*sin(k+f*x)^2)".ToEntity().Integrate("x")` | `integral(...)` | 71 characters |
+| `"(m+(1-m)*sec(c+d*x)^2)/sec(c+d*x)^m".ToEntity().Integrate("x")` | `integral(...)` | 56 characters |
+| `"(-(1-m)+(2-m)*cos(x)^2)/cos(x)^m".ToEntity().Integrate("x")` | `integral(...)` | 55 characters |
+| `"(3+3*sec(c+d*x)^2)/sec(c+d*x)^(1/2)".ToEntity().Integrate("x")` | `integral(...)` | 60 characters |
+| `"sqrt(sin(x))*(3/2-5/2*sin(x)^2)".ToEntity().Integrate("x")` | `integral(...)` | 27 characters |
 
 ### A power of a sine or hyperbolic sine over a binomial in a power of it is split over the binomial's roots
 

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

@@ -31310,8 +31310,8 @@ Piecewise piecewise when piecewise.Cases.Any() => piecewise.Cases.First().Expres
         }
 
         /// <summary>
-        /// <c>f^m (alpha + beta f^2)</c> for a symbolic <c>m</c> and <c>f</c> a sine, cosine, secant or
-        /// cosecant of a linear <c>u</c>, where it is the derivative of a product: with <c>u' = 1</c>,
+        /// <c>f^m (alpha + beta f^2)</c> for a symbolic <c>m</c>, or a number that is not whole, and <c>f</c> a
+        /// sine, cosine, secant or cosecant of a linear <c>u</c>, where it is the derivative of a product: with <c>u' = 1</c>,
         /// <c>(cos sin^(m + 1))' = sin^m ((m + 1) - (m + 2) sin^2)</c>,
         /// <c>(sin cos^(m + 1))' = cos^m ((m + 2) cos^2 - (m + 1))</c>,
         /// <c>(tan sec^m)' = sec^m ((m + 1) sec^2 - m)</c> and <c>(cot csc^m)' = -csc^m ((m + 1) csc^2 - m)</c>.
@@ -31321,23 +31321,39 @@ Piecewise piecewise when piecewise.Cases.Any() => piecewise.Cases.First().Expres
         /// square terms, <c>alpha + beta f^2</c>, which matches <c>p0 + p2 f^2</c> above where
         /// <c>beta p0 = alpha p2</c>; the answer is <c>(alpha/p0) G/u'</c>. Rubi's 4.1.4.1, 4.2.4.1 and 4.5.4.1
         /// <c>sec(c + d x)^m (m - (1 + m) sec(c + d x)^2)</c> and the like were declined.
+        /// The power is one of the integrand's factors, and one under the bar is <c>f^(-m)</c>: only then is the
+        /// integrand <c>f^m</c> times the rest, and a power in one term of a sum, as in
+        /// <c>-m sec^m + (1 + m) sec^2</c>, is not a factor of it.
         /// https://github.com/asc-community/AngouriMath/issues/718
         /// </remarks>
         internal static Entity? SolveASymbolicPowerOfATrigonometricFunctionAsTheDerivativeOfAProduct(Entity expr, Entity.Variable x)
         {
             if (!Integration.AnsweringTheQuestionAsked)
                 return null;
-            // The power with a symbol in its exponent, of a sine, cosine, secant or cosecant of a linear.
-            var power = expr.Nodes.OfType<Powf>().FirstOrDefault(node => node.Exponent.Vars.Any() && !node.Exponent.ContainsNode(x)
-                && node.Base is Sinf or Cosf or Secantf or Cosecantf);
-            if (power is not Powf(var function, var m))
-                return null;
+            // The power with a symbol in its exponent, of a sine, cosine, secant or cosecant of a linear,
+            // as a factor of the integrand.
+            var factors = FactorsOfTheIntegrand(expr);
+            var at = factors.FindIndex(pair => pair.Factor is Powf(Sinf or Cosf or Secantf or Cosecantf, var exponent)
+                && (exponent.Vars.Any() && !exponent.ContainsNode(x)
+                    // Or a number that is not whole: the derivatives hold for any m on the principal branch,
+                    // and a whole one is left to the rules for whole powers.
+                    || exponent is Number.Rational and not Number.Integer));
+            if (at < 0 || factors[at].Factor is not Powf(var function, var written))
+                return null;
+            var m = factors[at].Underneath ? (-written).InnerSimplified : written;
             var argument = function switch { Sinf(var y) => y, Cosf(var y) => y, Secantf(var y) => y, Cosecantf(var y) => y, _ => null };
             if (argument is null || !TreeAnalyzer.TryGetPolyLinear(argument, x, out var slope, out _) || Functions.PartialFractions.IsZeroAsAValue(slope))
                 return null;
             var s = Variable.CreateUnique(expr, "s_trig");
-            // The power taken off by writing 1 for it: `s^m (...)/s^m` is not cancelled by the simplification.
-            var inS = expr.Substitute(power, Number.Integer.One).Substitute(function, s).InnerSimplified;
+            // The rest of the factors, without the power: `s^m (...)/s^m` is not cancelled by the simplification.
+            Entity above = Number.Integer.One, below = Number.Integer.One;
+            for (var i = 0; i < factors.Count; i++)
+                if (i != at)
+                {
+                    if (factors[i].Underneath) below *= factors[i].Factor;
+                    else above *= factors[i].Factor;
+                }
+            var inS = (above / below).Substitute(function, s).InnerSimplified;
             if (inS.ContainsNode(x) || !TreeAnalyzer.TryGetPolynomial(inS.Expand(), s, out var read)
                 || read.Keys.Any(key => !key.IsZero && !key.Equals(EInteger.FromInt32(2))))
                 return null;

Sources/Tests/UnitTests/Calculus/ASymbolicPowerOfATrigonometricFunctionAsADerivativeIntegralTest.cs

@@ -24,7 +24,11 @@ private static void DifferentiatesBack(string integrand, params (string, double)
         {
             var integral = integrand.ToEntity().Integrate("x");
             Assert.DoesNotContain("integral(", integral.Stringize());
+            DifferentiatesBack(integrand, integral, pins);
+        }
 
+        private static void DifferentiatesBack(string integrand, Entity integral, params (string, double)[] pins)
+        {
             var pinned = integral.Substitute("C", 0);
             Entity original = integrand.ToEntity();
             foreach (var (name, value) in pins)
@@ -52,12 +56,46 @@ private static void DifferentiatesBack(string integrand, params (string, double)
         [InlineData("sin(c+d*x)^m*(1+m-(2+m)*sin(c+d*x)^2)")]
         [InlineData("cos(x)^m*(-(1+m)+(2+m)*cos(x)^2)")]
         [InlineData("csc(x)^m*(m-(1+m)*csc(x)^2)")]
+        // Under the bar the power is f^(-m): (tan sec^(-m))' and (sin cos^(1 - m))'.
+        [InlineData("(m+(1-m)*sec(c+d*x)^2)/sec(c+d*x)^m")]
+        [InlineData("(-(1-m)+(2-m)*cos(x)^2)/cos(x)^m")]
         public void IsAnsweredAndDifferentiatesBack(string integrand)
         {
             DifferentiatesBack(integrand, ("A", 0.6), ("q", 1.25), ("c", 0.3), ("d", 0.75), ("m", 2.5));
             DifferentiatesBack(integrand, ("A", -0.75), ("q", 0.4), ("c", -1), ("d", -0.5), ("m", 1.0 / 3));
         }
 
+        // A power that is not whole: Rubi's 4.5.4.1 (3 + 3 sec(c + d x)^2)/sqrt(sec(c + d x)), and the
+        // same derivatives at m = 1/2, 1/3 and 3/2.
+        [Theory]
+        [InlineData("(3+3*sec(c+d*x)^2)/sec(c+d*x)^(1/2)")]
+        [InlineData("sqrt(sin(x))*(3/2-5/2*sin(x)^2)")]
+        [InlineData("sin(c+d*x)^(1/3)*(4/3-7/3*sin(c+d*x)^2)")]
+        [InlineData("cos(x)^(3/2)*(-5/2+7/2*cos(x)^2)")]
+        [InlineData("csc(x)^(1/2)*(1/2-3/2*csc(x)^2)")]
+        public void IsAnsweredForAPowerThatIsNotWhole(string integrand)
+        {
+            DifferentiatesBack(integrand, ("c", 0.3), ("d", 0.75));
+            DifferentiatesBack(integrand, ("c", -1), ("d", -0.5));
+        }
+
+        // The power is not a factor of the whole integrand, or is one under the bar, where the bracket is
+        // not the derivative's: none of these is the derivative of a single product.
+        [Theory]
+        [InlineData("(-m+(1+m)*sec(x)^2)/sec(x)^m")]
+        [InlineData("-m*sec(x)^m+(1+m)*sec(x)^2")]
+        [InlineData("sin(x)^m*(1+m)-(2+m)*sin(x)^2")]
+        [InlineData("(1+m-(2+m)*sin(x)^2)/sin(x)^m")]
+        [InlineData("(-m+(1+m)*csc(x)^2)/csc(x)^m")]
+        public void IsNotAnsweredWronglyWhereThePowerIsNotThatFactor(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            if (integral.Stringize().Contains("integral("))
+                return;
+            DifferentiatesBack(integrand, integral, ("m", 2.5));
+            DifferentiatesBack(integrand, integral, ("m", 1.0 / 3));
+        }
+
         [Fact]
         public void IsNotClaimedWhereTheBracketIsNotTheDerivative()
         {