@@ -3100,6 +3100,24 @@ leaves are read as quotients of the sine and the cosine
 | `"1/(a+b*asin(c*x))".Integrate("x")` | `integral(1 / (a + b * arcsin(c * x)), x)` | `Si` and `Ci` of `(a + b arcsin(c x))/b` |
 | `"x/((c+a^2*c*x^2)^2*atan(a*x))".Integrate("x")` | `integral(x / ((c + a ^ 2 * c * x ^ 2) ^ 2 * arctan(a * x)), x)` | `1/2 * Si(2 * arctan(a * x)) / (a ^ 2 * c ^ 2) + C` |
 
+### The arcsecant beside a root of `x^2 - 1` is integrated right below `-1` too
+
+**Answers where there were none.** Under `x = sec(u)`, the substitution that undoes the arcsecant,
+`sqrt(x^2 - 1)` is `|tan(u)|`, and on the arcsecant's range the tangent has the sign of `x`. It was
+written `tan(u)`, so that on the unreleased master `x arcsec(x)/sqrt(x^2 - 1)` and its kin were
+answered right for `x > 1` and with the integrand's negative for a derivative at every `x < -1`,
+where they are as real; 2.5.0 declined them. The root is `tan(u) sgn(x)` now, as the cosecant's and
+the cotangent's carry their signs, and a first power of the arcsecant beside it is by parts. The sign
+written back is the argument's, which is that of `x` only for a positive slope and no constant:
+`arcsec(x + 3)` has the sign of `x + 3` ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x*asec(x)/sqrt(-1 + x^2)".ToEntity().Integrate("x")` | `integral(...)` | right for `x > 1` and for `x < -1` |
+| `"asec(x)/(x^2*sqrt(-1 + x^2))".ToEntity().Integrate("x")` | `integral(...)` | the same |
+| `"asec(x)^2/(x*sqrt(x^2 - 1))".ToEntity().Integrate("x")` | `integral(...)` | the same |
+| `"asec(x + 3)^2/((x + 3)*sqrt((x + 3)^2 - 1))".ToEntity().Integrate("x")` | `integral(...)` | right for `x > -2` and for `x < -4` |
+
 ### A perfect square in a power of the variable is read as one
 
 `sqrt(a^2 + 2 a b x^2 + b^2 x^4) sqrt(c + e x + d x^2)` was left unevaluated. The radicand is
 
@@ -7157,10 +7157,12 @@ private static bool TheInverseIsBelowTheBar(Entity expr, Entity inverse, Entity.
         /// and the cosine is not negative; substituting and simplifying instead would leave
         /// <c>sqrt(1 - sin(u)^2)</c>, which the simplifier is right not to call <c>cos(u)</c>.
         /// So <c>(1 - x^2)^(k/2)</c> becomes <c>cos(u)^k</c> directly, and likewise
-        /// <c>(1 + x^2)^(k/2)</c> into <c>sec(u)^k</c> for the tangent and <c>(x^2 - 1)^(k/2)</c>
-        /// into <c>tan(u)^k</c> for the secant, each on its principal branch. That branch is the
-        /// one the inverse function is defined on, so the answer holds wherever the integrand
-        /// is read through it.
+        /// <c>(1 + x^2)^(k/2)</c> into <c>sec(u)^k</c> for the tangent, each on its principal
+        /// branch. That branch is the one the inverse function is defined on, so the answer holds
+        /// wherever the integrand is read through it. For the secant <c>(x^2 - 1)^(k/2)</c> is
+        /// <c>|tan(u)|^k</c>, and on the arcsecant's range the tangent has the sign of the
+        /// argument, so it becomes <c>tan(u)^k</c> times that sign to the <c>k</c>, as the
+        /// cosecant's and the cotangent's roots carry theirs.
         /// </para>
         /// <para>
         /// Exactly one inverse function, of the variable or of a linear <c>c x + d</c> in it,
@@ -7213,9 +7215,10 @@ private static bool TheInverseIsBelowTheBar(Entity expr, Entity inverse, Entity.
             }
 
             var u = Variable.CreateUnique(expr, "u_inv");
-            // The sign of u, where the root's sign is that: a constant on each half of the
-            // range, carried through the integration as a symbol and written back as the sign
-            // of x, which it is for the cosecant and the cotangent.
+            // The sign the root has, where it has one: a constant on each half of the range,
+            // carried through the integration as a symbol and written back as the sign of the
+            // argument, which it is -- the sign of u for the cosecant and the cotangent, and of
+            // tan(u) for the secant.
             var signOfU = Variable.CreateUnique(expr, "s_inv");
             // x in terms of u, dx/du, the quadratic whose root goes by construction and what it
             // becomes, and the way back.
@@ -7245,7 +7248,11 @@ private static bool TheInverseIsBelowTheBar(Entity expr, Entity inverse, Entity.
                     (xInU, dxdu, radicandBase, root) = (MathS.Cosec(u), -MathS.Cosec(u) * MathS.Cotan(u), MathS.Sqr(argument) - 1, MathS.Cotan(u) * signOfU);
                     break;
                 default:
-                    (xInU, dxdu, radicandBase, root) = (MathS.Sec(u), MathS.Sec(u) * MathS.Tan(u), MathS.Sqr(argument) - 1, MathS.Tan(u));
+                    // arcsec has the range [0, pi/2) ∪ (pi/2, pi], where the tangent has the sign
+                    // of the argument: `sqrt(x^2 - 1)` is `tan(u) sgn(x)`. Taken as `tan(u)`, the
+                    // answer was right for x > 1 and its derivative the integrand's negative for
+                    // every x < -1, where the integrand is as real: `x arcsec(x)/sqrt(x^2 - 1)`.
+                    (xInU, dxdu, radicandBase, root) = (MathS.Sec(u), MathS.Sec(u) * MathS.Tan(u), MathS.Sqr(argument) - 1, MathS.Tan(u) * signOfU);
                     break;
             }
             if (linear)
@@ -7318,10 +7325,10 @@ node is Powf(var @base, Number.Integer power) && power.EInteger.CompareTo(EInteg
             {
                 if (radicalsRemoved == 0 && !exponentialOfTheInverse && !powerOfTheInverse)
                     return null;
-                // The cosecant and the cotangent carry the sign of u into the root, and a first
-                // power of either beside a root is parts' -- `arccot(x)/(1 + x^2)^(3/2)` is
-                // `x arccot(x)/sqrt(1 + x^2) + 1/sqrt(1 + x^2)` there, with no sign in it.
-                if (inverse is Arccosecantf or Arccotanf && !exponentialOfTheInverse && !powerOfTheInverse)
+                // The cosecant, the cotangent and the secant carry a sign into the root, and a
+                // first power of any of them beside a root is parts' -- `arccot(x)/(1 + x^2)^(3/2)`
+                // is `x arccot(x)/sqrt(1 + x^2) + 1/sqrt(1 + x^2)` there, with no sign in it.
+                if (inverse is Arccosecantf or Arccotanf or Arcsecantf && !exponentialOfTheInverse && !powerOfTheInverse)
                     return null;
             }
             // And nothing else of `x` under a root, which the construction did not reach:
@@ -7374,10 +7381,11 @@ node is Powf(var @base, Number.Rational power) && power is not Number.Integer &&
             if (result is null)
                 return null;
             if (result.ContainsNode(signOfU))
-                // The sign squared is one, and the sign itself is the sign of x.
+                // The sign squared is one, and the sign itself is the argument's: `arccsc(c x + d)`
+                // has the sign of `c x + d`, which is the sign of x only for a positive c and no d.
                 result = result
                     .Replace(node => node is Powf(var b, Number.Integer k) && b == signOfU ? (k.EInteger.IsEven ? Number.Integer.One : signOfU) : node)
-                    .Substitute(signOfU, MathS.Signum(x));
+                    .Substitute(signOfU, MathS.Signum(argument));
             return TrigonometryOfTheInverseInX(result.Substitute(u, inverse), inverse, argument);
         }
 
 
@@ -109,6 +109,26 @@ public void TheFourThatWereAlreadyThere(string integrand, double[] points)
         [InlineData("x^3*asec(x)/sqrt(x^4 - 1)", new[] { 1.4, 2.6, 5.1, -1.9, -3.7 })]
         public void ByPartsAgainstARootOfAQuartic(string integrand, double[] points) => DifferentiatesBack(integrand, points);
 
+        /// <summary>
+        /// The arcsecant beside a root of <c>x^2 - 1</c>, under <c>x = sec(u)</c>: the root is
+        /// <c>tan(u) sgn(x)</c> on the arcsecant's range, and taken as <c>tan(u)</c> the answers
+        /// were right for x &gt; 1 and their derivatives the integrands' negatives for every
+        /// x &lt; -1, where the integrands are as real. Timofeev's and Charlwood's. With a linear
+        /// argument the sign is the argument's: at <c>x = -1.5</c> the sign of <c>x + 3</c> is
+        /// not the sign of <c>x</c>.
+        /// </summary>
+        [Theory]
+        [InlineData("x*asec(x)/sqrt(-1 + x^2)", new[] { 1.3, 2.6, -1.3, -1.9, -2.7 })]
+        [InlineData("asec(x)*sqrt(-1 + x^2)/x^4", new[] { 1.3, 2.6, -1.3, -1.9, -2.7 })]
+        [InlineData("asec(x)/(-1 + x^2)^(5/2)", new[] { 1.3, 2.6, -1.3, -1.9, -2.7 })]
+        [InlineData("x^3*asec(x)/(-1 + x^2)^(5/2)", new[] { 1.3, 2.6, -1.3, -1.9, -2.7 })]
+        [InlineData("asec(x)/(x^2*sqrt(-1 + x^2))", new[] { 1.3, 2.6, -1.3, -1.9, -2.7 })]
+        [InlineData("(-1 + x^2)^(3/2)*asec(x)^2/x^5", new[] { 1.3, 2.6, -1.3, -1.9, -2.7 })]
+        [InlineData("asec(x)^3*sqrt(-1 + x^2)/x^4", new[] { 1.3, 2.6, -1.3, -1.9, -2.7 })]
+        [InlineData("asec(x)^2/(x*sqrt(x^2 - 1))", new[] { 1.3, 2.6, -1.3, -1.9, -2.7 })]
+        [InlineData("asec(x + 3)^2/((x + 3)*sqrt((x + 3)^2 - 1))", new[] { 0.5, -1.5, -0.5, -4.5, -6.0 })]
+        public void BesideARootOnBothIntervals(string integrand, double[] points) => DifferentiatesBack(integrand, points);
+
         /// <summary>
         /// A non-linear argument is not this row's: <c>arcsec(x^2)</c> would want the chain rule
         /// undone first, and the table reads a linear argument only.