[Rafael-SOWNet]

Part of #233.

• The derivative of a sum of exponentials over another, read off the integrand over the other's square. Rubi's 6.7.1 x (b - a sinh(x))/(a + b sinh(x))^2 ran past the budget, in three steps:

  1. the split took the bracket's terms apart;
  2. the substitution in e^x answered each over the square of a quadratic, with a piecewise on its discriminant;
  3. parts beside x integrated that piecewise again.

  For a sum D of exponentials of x, the integrand over D^2 is now read as (N/D)', where N is a constant plus a constant times each exponential in D.

• How N is found.

  • The exponentials' span is closed under the derivative, so N' D - N D' is linear in N's constants. With each exponential a whole power of the first, w, the identity read at six values of w is a linear system.
  • N and N + c D share a derivative over D. The solution with no constant is tried first, and that is -cosh(x)/(a + b sinh(x)).
  • Beside x, by parts, the integral of N/D is written as a constant times ln(D) where N is a constant times D', and asked as a question otherwise.

• Where it sits. It is the hyperbolic twin of SolveAsTheDerivativeOfASineRaisedFromNothingBesidePowersOfTheCosine, dispatched beside it. There are no hyperbolic nodes, since the parser writes sinh as exponentials.

Rubi row	master 78cd31b	this branch
6.7.1:643 x (b - a sinh(x))/(a + b sinh(x))^2	past 30 s	123 characters
6.7.1:644 x (b + a cosh(x))/(a + b cosh(x))^2	past 30 s	119 characters
6.7.1:1178 sech(x)^2 tanh(x)^2/(2 + tanh(x)^3)^2	117 characters, 16 s, 9.7 GB	-1/3/(2 + tanh(x)^3), 60 characters, 55 ms
(b - a sinh(x))/(a + b sinh(x))^2	7,861 characters, a piecewise	71 characters

Chain against master 78cd31b:

• The pocket of 2,652 rows with an exponential and a squared quotient went from 2,641 to 2,643 solved, with 0 wrong.
• Every family's sample is unchanged.
• The two rows that moved, 643 and 644, run alone, went from timeout to solved.
• 8,181 rows were compared for allocation, and none grew past 1.3x and 300 MB.

The full unit suite passes locally: 16,554 passed, 0 failed. The BREAKING-CHANGES lengths were measured on the branch. Rebased onto 37e2b1a.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura