Exponentials below the bar beside a power of a linear are integrated to the exponential integral

[Rafael-SOWNet]

Part of #718.

1/(x e^(2x)) was declined where e^(-2x)/x, the same function, was answered with Ei(-2x), and so was 1/((c + d x)(a + a tanh(e + f x))), Rubi's 6.3.1, and the same with coth, 6.4.1: the rules for an exponential over a linear read it above the bar only, and a hyperbolic function arrives as a quotient of exponentials. 2.5.0 declined them too:

integrand	2.5.0	master 752974bf	this
1/(x e^(2x))	declined	declined	Ei(-2x), in 0.6 s
1/(x (1 + tanh(x)))	declined	declined	ln(x)/2 + Ei(-2x)/2, in 0.2 s
1/((c + d x)(a + a tanh(e + f x)))	declined	declined	a logarithm and an exponential integral, in 0.3 s
1/((c + d x)^2 (a + a coth(e + f x))^3)	declined	declined	powers of c + d x and exponential integrals, in 0.4 s

What changes. Where exponentials of one linear stand below the bar beside a polynomial below it, they are written in the exponential every other one is a whole power of, w, and put over one bar; with the denominator in lowest terms a power of w alone, they are a sum of whole powers of w -- 1/(1 + tanh(z)) is (1 + e^(-2z))/2 -- and each term over the polynomial is the exponential integral's, as the rules above it answer it. Where anything else stays below the bar, 1/(x (1 + e^x)), the integral is neither elementary nor an exponential integral, and the rule declines at once.

Tests: ExponentialBelowTheBarIntegralTest.ExpandedOverTheLinear, eight rows, each differentiated back with the symbols pinned, on both sides of zero; all eight fail on master.

Measured first on the 1,082 problems of families 0, 2 and 6 with an exponential or a hyperbolic function and the variable below a bar outside it, 630 run, at the corpus's 5-second budget, against master 8f3757cd, the branch's base:

	master	this
solved	526	544
wrong	0	0
past the budget	11	11

Measured then on the Rubi corpus against master 8f3757cd:

	master	this
family 0, independent suites (1814)	1766	1769
family 1, 40 a file (1381)	1296	1297
families 2 to 8, sampled (2410)	2215	2234

No answer is wrong in the sample on either build. Of the 24 problems the two builds disagreed on, run again one build at a time, 19 are declined on master and answered here: the 14 of 6.3.1's and 6.4.1's in the sample, 1/((c + d x)^2 (a + a coth(e + f x))^3) among them; 2.3's 1/(F^(sqrt(1 - a x)/sqrt(1 + a x)) (1 - a^2 x^2)) and the same with twice the root; Apostol's 1/(e^t (t - 1 - a)), Timofeev's 1/(e^(x/2) x^3), and one of Hearn's. The other five are answered on both, after about twenty seconds on either build, past the corpus's budget.

The suite passes on the merge with master 752974bf, 0cd0ef2f, 14,681 tests with 13 skipped, run under a 4 GB heap limit, and the allocation gate on the commit measured, df2644c3; every row above is as the table says on the merge.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura