A root over a number times 1 + i tan(z) is integrated as it is over a + i a tan(z)

BREAKING-CHANGES.md

@@ -619,6 +619,21 @@ and the two together are a polynomial in `e^(i z)` and its reciprocal, multiplie
 | `"sqrt(tan(c + d*x))*(A + B*tan(c + d*x))/(a + i*a*tan(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | arctangents and logarithms of `sqrt(tan(c + d x))` |
 | `"(a + i*a*tan(c + d*x))^2*(A + B*tan(c + d*x))/(q - i*q*tan(c + d*x))^4".ToEntity().Integrate("x")` | `integral(...)` | a sum of multiples of `e^(k i (c + d x))` |
 
+### A root over a number times `1 + i tan(z)` is integrated as it is over `a + i a tan(z)`
+
+**Answers where there were none.** `sqrt(c + d tan(x))/(1 + i tan(x))` was declined, where
+`sqrt(c + d tan(x))/(a + i a tan(x))` is answered. Under the tangent `1 + i u` beside `1 + u^2` shares
+the root `u = i` with it, and the root was taken out first, as `i (u - i)^2 (u + i)`, only where a symbol
+stood below the bar: with numbers alone the split over the rationals and `i` was left to answer it,
+which it does for a polynomial above the bar and not for a root. The root is taken out beside anything
+above the bar that is not a polynomial now
+([#1788](https://github.com/asc-community/AngouriMath/issues/1788)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sqrt(c + d*tan(x))/(1 + i*tan(x))".ToEntity().Integrate("x")` | `integral(...)` | arctangents and logarithms of `sqrt(c + d tan(x))` |
+| `"(c + d*tan(x))^(3/2)/(1 + i*tan(x))".ToEntity().Integrate("x")` | `integral(...)` | the same, with a multiple of the root |
+
 ### A polynomial over a power of one linear with a symbol in it is written in powers of the linear
 
 **Answers where there were none.** `t^9/(a + b t)^8` is a polynomial and eight powers of `1/(a + b t)`,

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

@@ -782,7 +782,7 @@ is var (multiple, leftover)
             // `1/((a + i a tan(x)) (c + d tan(x)))`. The quadratic is written over the linears of
             // its roots and the shared one taken as one power, coprime and squarefree for the
             // splits below.
-            if (OverAComplexRootSharedWithAQuadratic(denominator, x) is { } overTheSharedRoot
+            if (OverAComplexRootSharedWithAQuadratic(numerator, denominator, x) is { } overTheSharedRoot
                 && (SolveByPartialFractions(numerator / overTheSharedRoot, x, integrateByParts)
                     ?? Integration.ComputeIndefiniteIntegral(numerator / overTheSharedRoot, x, integrateByParts)) is { } overTheComplexRoot)
                 return overTheComplexRoot;
@@ -14371,11 +14371,16 @@ private static bool IsASumOfMonomials(Entity expr, Entity.Variable x)
         /// where no quadratic shares such a root. <c>(1 + i x)(1 + x^2)</c> is
         /// <c>i (x - i)^2 (x + i)</c>.
         /// </summary>
-        private static Entity? OverAComplexRootSharedWithAQuadratic(Entity denominator, Entity.Variable x)
+        private static Entity? OverAComplexRootSharedWithAQuadratic(Entity numerator, Entity denominator, Entity.Variable x)
         {
-            // With numbers only, the split over the rationals and the imaginary unit answers it as
-            // it is written.
-            if (!denominator.Vars.Any(symbol => symbol != x))
+            // With numbers only below the bar and a polynomial above it, the split over the
+            // rationals and the imaginary unit answers it as it is written. Not with a root above
+            // it: `sqrt(c + d u)/((1 + i u)(1 + u^2))`, which the tangent substitution makes of
+            // `sqrt(c + d tan(x))/(1 + i tan(x))`, is a question for the substitution in the root,
+            // and in its variable the shared root is a quadratic beside a quartic, which nothing
+            // reads; with `a + i a u` for `1 + i u` it was answered, and with the number declined.
+            // https://github.com/asc-community/AngouriMath/issues/1788
+            if (!denominator.Vars.Any(symbol => symbol != x) && TreeAnalyzer.TryGetPolynomial(numerator, x, out _))
                 return null;
             var written = new List<(Entity Base, EInteger Power)>();
             Entity constant = Number.Integer.One;

Sources/Tests/UnitTests/Calculus/ComplexCoefficientRationalIntegralTest.cs

@@ -118,5 +118,19 @@ public void AnImaginaryMobiusPowerIsSettled(string integrand)
                 return;
             DifferentiatesBack(integrand);
         }
+
+        /// <summary>
+        /// A root above the bar and numbers below it: under the tangent <c>1 + i u</c> beside
+        /// <c>1 + u^2</c> shares the root <c>u = i</c> with it, and that was taken out only where a
+        /// symbol stood below the bar, so <c>a + i a tan(x)</c> was answered and
+        /// <c>1 + i tan(x)</c> declined.
+        /// https://github.com/asc-community/AngouriMath/issues/1788
+        /// </summary>
+        [Theory]
+        [InlineData("sqrt(c + d*tan(x))/(1 + i*tan(x))")]
+        [InlineData("sqrt(c + d*tan(x))/(2 + 2*i*tan(x))")]
+        [InlineData("sqrt(c + d*tan(x))/(1 - i*tan(x))")]
+        [InlineData("(c + d*tan(x))^(3/2)/(1 + i*tan(x))")]
+        public void ARootOverANumberTimesAnImaginaryTangentSum(string integrand) => DifferentiatesBack(integrand);
     }
 }