[Rafael-SOWNet]

Fifteen of Rubi's 4.7.6, 4.7.7, 6.1.4, 6.2.4 and 6.7.1 are exact derivatives written out, some of them beside an integral that is already answered. Five changes read them:

• The derivative of a product of powers may raise a sine, a cosine or an exponential from nothing, one at a time: F^(c (a + b x)) (f x)^m (e x cos(d + e x) + (m + b c x ln F) sin(d + e x))/x is (F^(c (a + b x)) (f x)^m sin(d + e x))'. A power with x in both its base and its exponent is read as its exponent's derivative times its base's logarithm, so cosh(x)^x (ln(cosh(x)) + x tanh(x)) is (cosh(x)^x)'.
• Where the brackets are not proportional, the integrand may be a constant times that derivative plus a constant times the functions alone: sinh(Q)/x^2 - b cosh(Q)/x is (-sinh(Q)/x)' + 2 c cosh(Q) for Q = a + b x + c x^2, and the second term is answered in error functions.
• A denominator with a negative exponential in it is multiplied above and below by the positive one, so (cos(x) + sin(x))/(e^(-x) + sin(x)) becomes the logarithmic derivative of 1 + e^x sin(x).
• cos(u)^2 + sin(u)^2 in a sum is written as 1.
• A polynomial factor that is a constant multiple of a substitution's du/dx is divided out: (x + b/(2 c)) tan(a + b x + c x^2) is -ln(cos(a + b x + c x^2))/(2 c).

Two more rows of 4.7.6, which the unreleased master answered in exponential integrals (1,209 and 2,331 characters), come out as the product F^(c (a + b x)) sin(d + e x)/x and its twin, at 50 and 53 characters.

Measured on 8a1ef454, branch against master:

Run	Problems	Master	Branch
Pocket: Rubi's 4.7, 6.1.4, 6.2.4 and 6.7	1,901	1,829 solved	1,844 solved
Family 1, 40 a file	1,381	1,362	1,362
Family 0, whole	1,814	1,789	1,789
Families 2 to 8, sampled	2,410	2,362	2,366

Re-run alone on both builds, 15 problems go from unevaluated to answered, in 15 ms to 2.5 s: Rubi's 4.7.6:53–55; 4.7.7:90, 977, 1180, 1199 and 1201; 6.1.4:14 and 19; 6.2.4:14 and 19; and 6.7.1:634, 766 and 1252. None goes the other way.

The sampled problem that master answered and the branch did not, 3.2.1:292, is one master answers in 23–32 s, against a 5 s budget. At a 60 s budget both builds answer it, and the allocations agree to 0.003% (34.2 GB). None of the problems answered on both builds allocates more on the branch. No run had a wrong answer.

AnExactDerivativeWithAFunctionRaisedFromNothingIntegralTest differentiates the answers back, and the 5,152 calculus tests pass. BREAKING-CHANGES.md has the entry, with the 2.5.0 column measured on v2.5.0.

Part of #233.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura