Take a factor constant on each side of zero for an inverse times its derivative

BREAKING-CHANGES.md

@@ -136,6 +136,22 @@ and the complex arcsine's real part moved two units away. A caller comparing a h
 result to a pinned string will see the last digits differ; a caller comparing to a tolerance
 will not. [#1338](https://github.com/asc-community/AngouriMath/issues/1338).
 
+### A function of an inverse sine of `sqrt(1 + b x^2)` beside its derivative keeps the sign of x
+
+**A wrong answer, and answers where there were none.** `1/(asin(sqrt(1 + b x^2)) sqrt(1 + b x^2))` was
+`sqrt(-b) ln(b asin(sqrt(1 + b x^2)))/b`, whose derivative is the integrand times the sign of x, so it was wrong for
+every negative x. `asin(sqrt(1 + b x^2))^n/sqrt(1 + b x^2)` was declined, and so were the arccosine twins of both
+(Rubi's 5.1.5 and 5.2.5). The ratio of the integrand to the inverse's derivative, `sqrt(-b x^2)/(b x)`, is constant
+on each side of 0 and changes sign across it. It is now compared on each side with that side's value, where it was
+compared everywhere with its value at one positive point
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"1/(asin(sqrt(1+b*x^2))*sqrt(1+b*x^2))".ToEntity().Integrate("x")` | `sqrt(-b) * ln(b * arcsin(sqrt(1 + b * x ^ 2))) / b + C`, wrong for x < 0 | 120 characters, `sqrt(1 - (1 + b x^2))/(b x) ln(arcsin(sqrt(1 + b x^2)))` |
+| `"asin(sqrt(1+b*x^2))^n/sqrt(1+b*x^2)".ToEntity().Integrate("x")` | `integral(...)` | 136 characters |
+| `"acos(sqrt(1+b*x^2))^n/sqrt(1+b*x^2)".ToEntity().Integrate("x")` | `integral(...)` | 137 characters |
+
 ### A rational evaluated at a higher precision than before has all of its digits
 
 **More digits where there were a hundred.** A rational kept its decimal value in the precision current when it was

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

@@ -29349,8 +29349,12 @@ static bool Through(Entity node, Entity.Variable x)
             // pinned: asking Simplify whether the derivative of the factor vanished took four seconds
             // on `sqrt(-a e^2/b - e^2 x^2)/(e sqrt(a + b x^2))`, and its square stayed a power of a
             // quotient of roots. A point it cannot tell at counts against it; the answer is checked too.
+            // Or on each side of 0 alone: `sqrt(-b x^2)/(b x)`, which `asin(sqrt(1 + b x^2))` beside
+            // `1/sqrt(1 + b x^2)` leaves, is constant where x keeps its sign and changes with it.
             bool HoldsItsValue(Entity factor)
-                => Functions.PartialFractions.HoldsAtSampledPoints(factor, factor.Substitute(x, Number.Rational.Create(3, 10)), x);
+                => Functions.PartialFractions.HoldsAtSampledPoints(factor, factor.Substitute(x, Number.Rational.Create(3, 10)), x)
+                || Functions.PartialFractions.HoldsAtSampledPoints(factor, factor.Substitute(x, Number.Rational.Create(3, 10)), x, new[] { "0.29", "1.43", "3.17" })
+                   && Functions.PartialFractions.HoldsAtSampledPoints(factor, factor.Substitute(x, Number.Rational.Create(-3, 10)), x, new[] { "-0.61", "-1.37", "-2.9" });
         }
 
         /// <summary>

Sources/Tests/UnitTests/Calculus/AFunctionOfAnInverseTimesItsDerivativeIntegralTest.cs

@@ -57,5 +57,15 @@ public void IsAnsweredAndDifferentiatesBack(string integrand)
             DifferentiatesBack(integrand, ("a", 9), ("b", -1.25), ("c", 0.7), ("d", 1.4), ("k", 0.75), ("m", 2.5));
             DifferentiatesBack(integrand, ("a", 7), ("b", -2), ("c", -0.5), ("d", 0.6), ("k", -0.3), ("m", 1.0 / 3));
         }
+
+        // A factor constant on each side of 0 and not across it: sqrt(-b x^2)/(b x).
+        [Theory]
+        [InlineData("asin(sqrt(1+b*x^2))^m/sqrt(1+b*x^2)")]
+        [InlineData("1/(acos(sqrt(1+b*x^2))*sqrt(1+b*x^2))")]
+        public void AFactorConstantOnEachSideOfZero(string integrand)
+        {
+            DifferentiatesBack(integrand, ("b", -0.1), ("m", 2.5));
+            DifferentiatesBack(integrand, ("b", -0.15), ("m", 1.0 / 3));
+        }
     }
 }