A power of an exponential over a polynomial is integrated to the exponential integral

[Rafael-SOWNet]

Part of #718.

(F^(g (e + f x)))^n with a symbol for F or n was read by no rule below a polynomial: (F^x)^n/x, and Rubi's 2.2, (a + b (F^(g (e + f x)))^n)^p/(c + d x)^m, were declined, while F^(g (e + f x))/(c + d x) is answered with Ei. 2.5.0 declined them too:

integrand	2.5.0	master ac6e33fa	this
(F^x)^2/x	declined	declined	Ei(2 ln(F) x), in 0.3 s
(F^x)^n/x	declined	declined	(F^x)^n e^(-n ln(F) x) Ei(n ln(F) x), in 0.1 s
(a + b (F^(g (e + f x)))^n)/(c + d x)	declined	declined	a logarithm and an exponential integral, in 0.2 s
(a + b (F^(g (e + f x)))^n)^3/(c + d x)^3	declined	declined	powers of c + d x and exponential integrals, in 0.3 s

What changes. The power has the constant logarithmic derivative n g f ln F, so it is a constant times e^(n g f ln(F) x) wherever it is differentiable. With a symbol for that constant, beside a polynomial below the bar, the integrand is one the exponential integral's rules read, and the answer is written back with the constant as (F^(g (e + f x)))^n e^(-n g f ln(F) x), whose derivative is zero. That holds for every F: for a negative one, (F^x)^n differs from F^(n x) by a factor that changes wherever F^x crosses the negative axis, and the constant carries it, where writing the power as F^(n g (e + f x)) would be right only for a positive F. A whole power is that exactly, whatever the base, and is written so.

Tests: PowerOfAnExponentialIntegralTest, six rows over a linear differentiated back with the symbols pinned and F = 3, and two with F = -3 compared as complex numbers on both sides of the axis crossings; all eight fail on master.

Measured first on the 999 problems of family 2 and the others with a power of an exponential in them, 675 run, at the corpus's 5-second budget, against master 8f3757cd, the branch's base:

	master	this
solved	596	606
wrong	0	0
past the budget	5	5

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	1297
families 2 to 8, sampled (2410)	2215	2228

No answer is wrong in the sample on either build. Of the 15 problems the two builds disagreed on, run again one build at a time, 10 are declined on master and answered here: the nine of 2.2's (a + b (F^(g (e + f x)))^n)^p/(c + d x)^m in the sample, and 2.3's (F^(sqrt(1 - a x)/sqrt(1 + a x)))^n/(1 - a^2 x^2), which master declined after 7 s and is answered in a tenth of one. The other five are answered on both, after about twenty seconds on either build, past the corpus's budget.

The suite passes on the commit measured, 6cb29bb6, 14,599 tests with 13 skipped, and the allocation gate with it. On the merge with master ac6e33fa, 3e3e9567, the 4,060 calculus and corpus tests pass, and every row of the first table is as it says.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura