[Rafael-SOWNet]

Part of #233.

• Half-odd powers of a hyperbolic sine beside an exponential, read as a derivative. Rubi's 6.7.1 e^(a + b x) sinh(c + d x)^(3/2) - 3/4 e^(a + b x) d^2/((b^2 - 9/4 d^2) sqrt(sinh(c + d x))) was declined. Each term alone is not elementary, and the sum is the derivative of e^(a + b x) (4 b sinh^(3/2) - 6 d cosh sinh^(1/2))/(4 b^2 - 9 d^2).
• The ansatz. Write E = e^(a + b x), S = sinh(u), C = cosh(u) and u = c + d x. Then (E S^p C^q)' = E (b S^p C^q + d p S^(p - 1) C^(q + 1) + d q S^(p + 1) C^(q - 1)), with C^2 = 1 + S^2. So the answer is a sum of E S^p C^q over the integrand's monomials and the partners their derivative reaches, and its coefficients are a linear system.
• Solving it. The system is square, one row per monomial of the ansatz. Each compound coefficient of the integrand is named while it is solved. Without that, the symbolic elimination reduced a row to 0 = c, where c is zero only as a value, and declined.
• Branches. Powers of one sine multiply on every branch, and no power of a power is taken. The rule is last in the dispatch, and the answer is checked by its derivative.

Input	master ef4150e	this branch
6.7.1:1141	declined	249 characters, e^(a + b x) (4 b S^(3/2) - 6 d S^(1/2) C)/(4 b^2 - 9 d^2)

Chain against master ef4150e (the branch rebased onto 7d1081e):

• The family-6 pocket of 3,690 rows went from 3,676 to 3,681 solved.
• Of the five rows that moved, run alone, 6.7.1:1141 is this rule's. The other four, 6.5.3:251/255 and 6.6.3:213/217, are Read x times a rational function of an exponential of a logarithm off the integrand #2047's, merged since the master arm was cut.
• Every family's sample is unchanged.
• 9,281 rows were compared for allocation, and none grew past 1.3x and 300 MB.

The library builds for netstandard2.0, net8.0 and net10.0. The suite is left to CI. The BREAKING-CHANGES length was measured on the branch, and the 2.5.0 column on a v2.5.0 build.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura