The scaling rule does not simplify an integrand with the imaginary unit in it

BREAKING-CHANGES.md

@@ -465,6 +465,24 @@ interval between the poles of the tangent, as the substitution's are
 | `"(A + B*tan(x))/(a + i*a*tan(x))^2".ToEntity().Integrate("x")` | `integral(...)` | the same |
 | `"1/(a + i*a*tan(c + d*x))^3".ToEntity().Integrate("x")` | `integral(...)` | the same in `tan(c + d x)` |
 
+### An integrand with the imaginary unit in it is not simplified by the rule that scales the variable
+
+**Answers where there were none.** The rule that scales the variable by the integrand's one symbol
+simplified the scaled integrand three times, to separate the whole power of the scale it is
+homogeneous of. With the imaginary unit beside the symbol that search ran past any budget: on the
+unreleased master `((1 + i a x)/(1 - i a x))^(-5/4)/x^2` took more than two minutes to be declined,
+and the rules after this one never had their turn on integrands that it held up. Such an integrand is
+inner-simplified now, and declined in seconds where the parts do not separate
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sec(c + d*x)^3/(a + i*a*tan(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | logarithms of `tan((c + d x)/2) ± 1` and a quotient |
+| `"tan(c + d*x)^(8/3)/(a + i*a*tan(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | an antiderivative in `tan(c + d x)^(1/3)` |
+| `"1/(a + b*csc(c + d*x)^2)^4".ToEntity().Integrate("x")` | `integral(...)` | an antiderivative, by the signs of `a` and `b` |
+| `"x^4*e^(2*i*atan(a + b*x))".ToEntity().Integrate("x")` | `integral(...)` | a polynomial, logarithms and arctangents of `a + b x` |
+| `"sec(c + d*x)^2/(a + i*a*tan(c + d*x))".ToEntity().Integrate("x")` | `ln(i a d tan(c + d x) + a d)/(i a d)` | `ln(i tan(c + d x) + 1)/(i a d)`, the same up to a constant |
+
 ### `NaN` was returned as the antiderivative of something that has one
 
 **A wrong answer, not a missing one.** `1/(a*x^2)` came back as `NaN + C`, and `NaN` is this

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

@@ -18538,7 +18538,13 @@ bool IsAConstantTimesAPowerOfOneMinusUSquared(Entity polynomial, out Entity cons
 
             // dx = c dt.
             var t = Variable.CreateUnique(expr, "u_scale");
-            var scaled = (expr.Substitute(x, scale * t) * scale).Simplify();
+            // With the imaginary unit in the integrand the simplifier's search beside the symbol
+            // does not end within any budget, and inner simplification separates what a whole
+            // power of the scale does: `((1 + i a x)/(1 - i a x))^(-5/4)/x^2` under its radical's
+            // substitution was minutes here before it was declined. Where the parts do not
+            // separate so, the check below declines it.
+            Entity Simplified(Entity e) => HoldsTheImaginaryUnit(expr) ? Functions.PartialFractions.Bare(e.InnerSimplified) : e.Simplify();
+            var scaled = Simplified(expr.Substitute(x, scale * t) * scale);
             if (scaled.ContainsNode(x))
                 return null;
 
@@ -18547,11 +18553,11 @@ bool IsAConstantTimesAPowerOfOneMinusUSquared(Entity polynomial, out Entity cons
             // The integrand with the scale set to one is the candidate h(t), and what is left
             // over when the scaled integrand is divided by it is the candidate factor. Read off
             // rather than searched for: if the two do separate, the quotient *is* the factor.
-            var withoutScale = scaled.Substitute(scale, Number.Integer.One).Simplify();
+            var withoutScale = Simplified(scaled.Substitute(scale, Number.Integer.One));
             if (withoutScale.ContainsNode(scale) || withoutScale == Number.Integer.Zero)
                 return null;
 
-            var factor = (scaled / withoutScale).Simplify();
+            var factor = Simplified(scaled / withoutScale);
             // Collapsing the quotient attaches the condition that the denominator it cleared is
             // non-zero. That denominator is the integrand's own, so the condition says where the
             // integrand is defined and nothing about this rewrite; it is dropped for the same

Sources/Tests/UnitTests/Calculus/ComplexCoefficientRationalIntegralTest.cs

@@ -79,5 +79,20 @@ private static void DifferentiatesBack(string integrand)
         [InlineData("(A + B*tan(x))/(a + i*a*tan(x))^2")]
         [InlineData("1/(a + i*a*tan(c + d*x))^3")]
         public void APowerOfAPlusIATangent(string integrand) => DifferentiatesBack(integrand);
+
+        /// <summary>
+        /// A power of a quotient of linears with the imaginary unit in it, beside a power of x the
+        /// substitution for it does not take down to something answered: declined, and within a
+        /// minute. The scaling of the variable simplified it with the symbol beside the imaginary
+        /// unit, and that ran past two minutes before the decline.
+        /// </summary>
+        [Theory]
+        [InlineData("((1 + i*a*x)/(1 - i*a*x))^(-5/4)/x^2")]
+        [InlineData("((1 + i*a*x)/(1 - i*a*x))^(-5/4)/x^3")]
+        public void AnImaginaryMobiusPowerIsDeclinedWithinAMinute(string integrand)
+        {
+            var integrating = System.Threading.Tasks.Task.Run(() => integrand.ToEntity().Integrate("x"));
+            Assert.True(integrating.Wait(TimeSpan.FromSeconds(60)), $"{integrand} was not settled within a minute");
+        }
     }
 }