[Rafael-SOWNet]

Part of #233.

Five shapes of Rubi's 6.1.1 and 6.7.1, each answered through an identity of the hyperbolic functions that the substitutions did not use. One test class per shape.

1. (g + f x)^m cosh(c + d x)^k/(a + i a sinh(c + d x)) and its sinh^k rows. Since cosh^2 = (1 + i sinh)(1 - i sinh), the numerator divides by the denominator, leaving a polynomial and a constant over it.
2. 1/(sech(x) ± i tanh(x))^4 is cosh(x)^4/(1 ± i sinh(x))^4, which the same identity cancels to (1 ∓ i sinh)^2/(1 ± i sinh)^2. It applies only to cosh^(2m) over (1 ± i sinh)^(2m) and the other way up; cancelling where something else is left made other answers longer.
3. A symbolic power of sinh, cosh, sech or csch beside an odd power of the other, under u = that function. A power of csch stays a power of u, since it is not sinh^(-n) where the sine is negative.
4. (cosh(x) coth(x))^(3/2) and (csch(x) + sinh(x))^(3/2) are both (cosh(x)^2/sinh(x))^(3/2), which is cosh^3 sinh^(-3/2) times a constant on each side of zero.
5. Half-odd powers of S = b cosh(x) + c sinh(x) ± sqrt(b^2 - c^2). With T = b sinh + c cosh, d/dx (T S^(n-1)) = n S^n - a (2n - 1) S^(n-1) steps the power to ±1/2. The branch first also stepped whole powers up to T/(a S). On today's master the exponential substitution writes 1/S^2, 1/S^3 and 1/S^4 in 182, 272 and 416 characters, where the recurrence gave 350, 539 and 728, so whole powers are now left to it.

Measured.

• Chain against master 4877ed2, the branch's base, with the harness at its current revision on both arms:
  • pocket of 1,756 rows: 1,686 → 1,723 solved, 0 wrong;
  • samples: unchanged or better;
  • the 43 rows that moved, re-run alone: 4 → 42 solved.
• Re-measured with the probe on master f7f2a7a, after the rebase. 14 of those 43 rows are new there:
  • twelve rows of (e + f x)^k cosh^m/(a + i a sinh) go from past 20 s to 0.1–0.9 s;
  • (csch(x) + sinh(x))^(3/2) and ^(5/2) go from declined to answered.
• The other 29 answer on both builds, most of them much cheaper here:

input	master f7f2a7a	this
sech(a+b*x)^n*tanh(a+b*x)^3	11.2 s, 9.12 GB, 180 characters	71 ms, 0.02 GB, 183
cosh(a+b*x)^3*sinh(a+b*x)^n	1.16 GB, 326	0.02 GB, 149
1/(sech(x)+i*tanh(x))^4	13.5 s, 10.5 GB, 551	0.08 GB, 241
(sech(x)+i*tanh(x))^4	2,916	251
(g+f*x)*cosh(c+d*x)^2/(a+i*a*sinh(c+d*x))	26.9 s, 27.5 GB, 740	0.07 GB, 366
1/sqrt(b*cosh(x)+c*sinh(x)+sqrt(b^2-c^2))	0.42 GB, 467	0.004 GB, 409

Every answer was checked by differentiating back at the probe's points; the test classes pin each shape the same way. 426 tests pass locally across the five classes, the hyperbolic tests and the corpus gate.

The BREAKING-CHANGES.md entry has 2.5.0's column measured. Its "Now" column and its comparison with the unreleased master are re-measured on this build and on f7f2a7a.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura