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Write each linear once under a symbolic power #1974

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Rafael-SOWNet merged 1 commit into from
Oct 20, 2026
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Write each linear once under a symbolic power #1974

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20 changes: 20 additions & 0 deletions BREAKING-CHANGES.md
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Expand Up @@ -4763,6 +4763,26 @@ and this is the same answer reached through the even root
|---|---|---|
| `"sqrt(x)/(sqrt(1/x + 1)*sqrt(1 - x^2))".Integrate("x")` | `integral(...)` | an antiderivative in `sqrt(x)`, `provided x >= 0` |

### A product of powers of quotients of linears with a symbolic exponent is written over each linear once

**Answers where there were none.** `e^atanh(a x) x^m sqrt(c - a^2 c x^2)`, `e^(2 p atanh(a x)) (c - a^2 c x^2)^p` and
seventeen more of Rubi's 7.3.6 and 7.4.2 were declined. The first is `sqrt((1 + a x)/(1 - a x)) x^m sqrt(c - a^2 c x^2)`,
in which the root `1/a` stands in both bases with exponents `-1/2` and `1/2`, and the second is
`K (a x + 1)^(2 p)`; the rule below reads rational exponents only. Each exponent of a linear is now the sum, symbols
and all, of those it has in the bases, and `K`, the integrand over that product, has a logarithmic derivative of zero
on every branch, so it stands in front of the answer. Only where the product comes to one power that is not whole, of
x or of a linear, beside whole powers of the other linears above the bar: the antiderivative is then a sum of powers
of it on every interval between the roots, and is checked at sampled points. Two powers that are not whole are left
to the rules that answer them as a power of the quotient
([#233](https://github.com/asc-community/AngouriMath/issues/233)).

| Input | Was (2.5.0) | Now |
|---|---|---|
| `"e^atanh(a*x)*x^m*sqrt(1-a^2*x^2)".ToEntity().Integrate("x")` | `integral(...)` | `K (x^(m + 1)/(m + 1) + a x^(m + 2)/(m + 2))`, 151 characters |
| `"e^atanh(a*x)*x^m*(c-a^2*c*x^2)^(5/2)".ToEntity().Integrate("x")` | `integral(...)` | 619 characters |
| `"e^(2*p*atanh(a*x))*(c-a^2*c*x^2)^p".ToEntity().Integrate("x")` | `integral(...)` | `K (a x + 1)^(2 p + 1)/(a (2 p + 1))`, 153 characters |
| `"e^(n*acoth(a*x))*(c-a*c*x)^(2+1/2*n)".ToEntity().Integrate("x")` | `integral(...)` | 585 characters |

### A product of powers of quotients of linears is written over each linear once

**Improvement, not silent.** `e^atanh(a x) sqrt(c - c/(a x))` is
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