Undo an inverse sine or cosine to a power that is not whole

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

@@ -12882,9 +12882,15 @@ private static bool TheInverseIsBelowTheBar(Entity expr, Entity inverse, Entity.
             // leaves `x^2 arccsc(x)/sqrt(x^2 - 1)` and stalls, where under `x = csc(u)` it is
             // `-u^2 csc(u)^4 cot(u)`, two steps of parts against a power of the cosecant.
             // Or of a function of it alone -- `(a + b arcsin(c x))^2` -- which is a power of
-            // u after the substitution.
+            // u after the substitution. Or, of an inverse sine or cosine, a power that is not whole,
+            // which parts cannot lower: `x^2 arcsin(a x)^n` under the sine is
+            // `u^n sin(u)^2 cos(u)/a^3`, Euler's formula's exponentials of `i u` beside a power of u,
+            // each a gamma function. Not of the others: `1/((c + a^2 c x^2) sqrt(arctan(a x)))` is
+            // a derivative of the inverse's root, answered in 0.7 s elsewhere and in 5 s under the
+            // tangent.
             var powerOfTheInverse = expr.Nodes.Any(node =>
-                node is Powf(var @base, Number.Integer power) && power.EInteger.CompareTo(EInteger.One) > 0
+                node is Powf(var @base, var power)
+                    && (power is Number.Integer whole ? whole.EInteger.CompareTo(EInteger.One) > 0 : inverse is Arcsinf or Arccosf && !power.ContainsNode(x))
                     && @base.ContainsNode(inverse) && !@base.Substitute(inverse, u).ContainsNode(x));
             // Or, asked for the sine and cosine integrals, the inverse function below the bar, or
             // a function of it alone there: `x^m/(a + b arcsin(c x))^n` is a polynomial in the

Sources/Tests/UnitTests/Calculus/APowerBesideAnExponentialOfAPowerIntegralTest.cs

@@ -205,6 +205,19 @@ public void APositiveRateIsTakenOutOfThePower()
         [InlineData("x^2*e^(b*(1 + 4*x + 6*x^2 + 4*x^3 + x^4))")]
         public void AnExponentThatIsAPowerOfAShiftedLinearWrittenExpanded(string integrand) => DifferentiatesBack(integrand);
 
+        /// <summary>
+        /// A power that is not whole of an inverse sine or cosine, which the substitution that undoes it writes as a power of
+        /// u beside a polynomial in the sine and cosine of u, Euler's formula's exponentials of <c>i u</c>, each a gamma
+        /// function. Parts cannot lower such a power, and the substitution was taken only for a whole one. Rubi's 5.1.2,
+        /// 5.2.2 and 5.1.5.
+        /// </summary>
+        [Theory]
+        [InlineData("asin(a*x)^n")]
+        [InlineData("x*acos(a*x)^n")]
+        [InlineData("x^2*(d + f*asin(a*x))^n")]
+        [InlineData("(c + a*asin(d - a*x))^n")]
+        public void ASymbolicPowerOfAnInverseSineOrCosine(string integrand) => DifferentiatesBack(integrand);
+
         /// <summary>
         /// Not where the integral is elementary, an exponential integral or an error function: those keep the form
         /// they had.