An exponential of a hyperbolic sine or cosine beside a function of it over its derivative is integrated

[Rafael-SOWNet]

Part of #718.

e^(n cosh(a + b x)) tanh(a + b x) is Ei(n cosh(a + b x))/b, as e^(n cos(x)) tan(x) is -Ei(n cos(x)), which is answered; the hyperbolic one was declined after seconds. A hyperbolic function arrives as exponentials, and the substitution that reads the sine or cosine as a node has none to read. 2.5.0 declined them too:

integrand	2.5.0	master 9d40ce3b	this
e^(n cosh(a + b x)) tanh(a + b x)	declined after 3 s	declined after 7 s	Ei(n cosh(a + b x))/b, in 0.4 s
e^(n sinh(a + b x)) coth(a + b x)	declined after 1 s	declined after 4 s	Ei(n sinh(a + b x))/b, in 0.05 s
e^(n sinh(a + b x)) sinh(2 (a + b x))	declined after 15 s	declined after 4 s	elementary, in 0.04 s
e^(n cosh((a + b x)/2)) sinh(a + b x)	declined after 9 s	declined after 4 s	elementary, in 0.04 s

What changes. Written in w = e^L, the integrand beside the outer exponential, over the derivative of the hyperbolic sine or cosine in its exponent, is asked as a rational function of u = w + s/w, twice that sine or cosine. Where it is one, it is integrated in u, the exponential of n u/2 beside it, and written back with u = e^L + s e^(-L). Rubi's 6.7.1.

Tests: ExponentialOfAHyperbolicIntegralTest, five rows differentiated back with the symbols pinned, on both sides of zero; all five are declined on master.

Measured first on the 20 problems of 6.7.1 with an exponential of a hyperbolic sine or cosine in them, at the corpus's 5-second budget, against master 8f3757cd, the branch's base:

	master	this
solved	2	20
wrong	0	0
past the budget	0	0

Measured then on the Rubi corpus against master 8f3757cd:

	master	this
family 0, independent suites (1814)	1766	1766
family 1, 40 a file (1381)	1296	1296
families 2 to 8, sampled (2410)	2215	2219

No answer is wrong in the pocket or the sample on either build. Of the 23 problems the two builds disagreed on, pocket and sample together, run again one build at a time, master answers 5 and this all 23: eighteen of 6.7.1, declined on master after one to five seconds and answered here in hundredths of one, and five that take about twenty seconds on either build, at the edge of the harness's patience, and are answered by both run alone.

The suite passes on the commit measured, ada35204, 14,596 tests with 13 skipped, and the allocation gate with it. On the merge with master 346b2181, 39198cd7, the 4,082 calculus and corpus tests that run pass, with 2 skipped. Every row of the first table is as it says on the merge.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura