A quadratic divided through by its leading coefficient is read as a multiple of the radicand

BREAKING-CHANGES.md

@@ -1790,6 +1790,19 @@ to a constant ([#1501](https://github.com/asc-community/AngouriMath/issues/1501)
 | `"sinh(x)^2/x".Integrate("x")` | `integral(((e ^ x - e ^ (-x)) / 2) ^ 2 / x, x)` | `-1/2 * ln(x) + 1/2 * Chi(2 * x) + C` |
 | `"cosh(a+b*x)/(c+d*x)".Integrate("x")` | `integral((e ^ (a + b * x) + e ^ (-(a + b * x))) / 2 / (c + d * x), x)` | `Chi` and `Shi` of `b (c + d x)/d`, with `cosh(a - b c/d)/d` and `sinh(a - b c/d)/d` in front, written in exponentials |
 
+### An inverse hyperbolic tangent over a power of `1 - a^2 x^2` is integrated to the hyperbolic sine and cosine integrals
+
+`x/((1 - a^2 x^2)^2 artanh(a x))` ran out of time. Under `x = tanh(u)/a` it is
+`sinh(u) cosh(u)/(a^2 u)`, but the quadratic reached the substitution divided through by its
+leading coefficient, as `1/(-a^2) + x^2`, and was not recognised as a multiple of `1 - a^2 x^2`. It
+is now, and these integrate to `Shi` and `Chi` of `2 artanh(a x)`
+([#1501](https://github.com/asc-community/AngouriMath/issues/1501)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x/((1-a^2*x^2)^2*atanh(a*x))".Integrate("x")` | `integral(x / ((1 - a ^ 2 * x ^ 2) ^ 2 * 1/2 * ln((1 + a * x) / (1 - a * x))), x)` | `Shi(2 artanh(a x))/(2 a^2)`, with `artanh` written as its logarithm |
+| `"1/((1-a^2*x^2)^2*atanh(a*x))".Integrate("x")` | `integral(1 / ((1 - a ^ 2 * x ^ 2) ^ 2 * 1/2 * ln((1 + a * x) / (1 - a * x))), x)` | `(ln(artanh(a x)) + Chi(2 artanh(a x)))/(2 a)`, likewise |
+
 ### A definite integral is the limit at a bound where its antiderivative is undefined
 
 `integral(x * e^(-x), x, 0, +oo)` was `NaN`, this library's way of saying a value does not

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

@@ -5586,6 +5586,14 @@ node is Powf(var @base, Number.Rational power) && power is not Number.Integer &&
             if (!AreProportionalAtSampledPoints(candidate, wanted, x))
                 return null;
             var ratio = Functions.PartialFractions.Bare((candidate / wanted).Simplify());
+            // Two proportional quadratics whose quotient does not simplify free of x: the
+            // integrand's `1 - a^2 x^2` arrives divided through by its leading coefficient, as
+            // `1/(-a^2) + x^2`, and the ratio is that of the leading coefficients.
+            if (ratio.ContainsNode(x)
+                && TreeAnalyzer.TryGetPolyQuadratic(candidate, x, out var candidateLeading, out _, out _)
+                && TreeAnalyzer.TryGetPolyQuadratic(wanted, x, out var wantedLeading, out _, out _)
+                && !TreeAnalyzer.IsZero(wantedLeading))
+                ratio = Functions.PartialFractions.Bare((candidateLeading / wantedLeading).Simplify());
             if (ratio.ContainsNode(x) || ratio.Nodes.Any(node => node == MathS.NaN))
                 return null;
             // A multiple that is known negative is refused: `sqrt(k Q)` is `sqrt(k) sqrt(Q)`

Sources/Tests/UnitTests/Calculus/HyperbolicIntegralIntegrationTest.cs

@@ -67,6 +67,21 @@ private static readonly (string, string)[] Pins =
         public void ASumOfExponentialsOverAPowerOfALinear(string integrand)
             => DifferentiatesBack(integrand, AroundZero, Pins);
 
+        /// <summary>
+        /// An inverse hyperbolic tangent below the bar, beside a whole power of <c>1 - a^2 x^2</c>:
+        /// under <c>x = tanh(u)/a</c> it is a polynomial in <c>sinh(u)</c> and <c>cosh(u)</c> over
+        /// a power of <c>u</c>. The quadratic reaches the substitution divided through by its
+        /// leading coefficient, as <c>1/(-a^2) + x^2</c>, and is read as a multiple of
+        /// <c>1 - a^2 x^2</c> by the ratio of the two leading coefficients. Rubi's 7.3.4,
+        /// <c>x^m/((1 - a^2 x^2)^k artanh(a x)^p)</c>.
+        /// </summary>
+        [Theory]
+        [InlineData("x/((1 - a^2*x^2)^2*atanh(a*x))")]
+        [InlineData("1/((1 - a^2*x^2)^2*atanh(a*x))")]
+        [InlineData("x^2/((1 - a^2*x^2)^3*atanh(a*x)^2)")]
+        public void AnInverseHyperbolicTangentBelowTheBar(string integrand)
+            => DifferentiatesBack(integrand, AroundZero, Pins);
+
         [Theory]
         [InlineData("sinh(x)/x", "Shi(x)")]
         [InlineData("cosh(x)/x", "Chi(x)")]

Sources/Tests/UnitTests/Calculus/TrigonometricIntegralIntegrationTest.cs

@@ -135,6 +135,7 @@ public void UnderAPowerOfTheVariable(string integrand)
         [InlineData("1/acos(x)")]
         [InlineData("sqrt(1 - x^2)/asin(x)")]
         [InlineData("x/((c + d^2*c*x^2)^2*atan(d*x))")]
+        [InlineData("1/((c + d^2*c*x^2)^3*atan(d*x))")]
         public void ThroughAnInverseTrigonometricFunction(string integrand)
             => DifferentiatesBack(integrand, new[] { 0.15, 0.3, 0.5, 0.7 }, Pins);