e^(n i arctan(a x)) to a power that is not whole is one power of a quotient of linears

[Rafael-SOWNet]

Part of #718.

e^(n i arctan(L)) is written algebraically, as (1 + i L)^n (1 + L^2)^(-n/2), before it is integrated, and for an n that is not whole those are two radicals of different orders that nothing reads: every power of e^(i arctan(a x)) that is not whole was declined, which is most of Rubi's 5.3.6.

integrand	master 3a3d0163	this
x^2 e^(3/2 i arctan(a x))	declined	2.5 s
e^(3/2 i arctan(a x))/x^4	declined	2.5 s
x^3 e^(-3/2 i arctan(a x))	declined	1.2 s
x e^(1/3 i arctan(x))	declined	0.3 s

What changes. For an n that is not whole, e^(n i arctan(L)) is written ((1 + i L)/(1 - i L))^(n/2). For a real L that is the same on the principal branch: the quotient is e^(2 i arctan(L)), and 2 arctan(L) is its principal argument. It is one power of a quotient of linears, which the substitution for such a power reads. A whole n keeps the form it had.

Tests: InverseTrigonometricSubstitutionTest.ToAPowerThatIsNotWhole, five rows, each differentiated back.

Measured first on the 257 problems of 5.3.6 with an exponential of i times an arctangent in them, at the corpus's 5-second budget, against master 9c2509d8, the branch's base:

	master	this
solved	123	207
wrong	0	0

85 more are answered. One master answers, x^3/e^(3 i arctan(a + b x)), timed out here; its n is whole and its path unchanged, and unbudgeted both builds answer it in 12 s.

Measured then on the Rubi corpus against master 9c2509d8:

	master	this
family 0, independent suites (1814)	1766	1766
family 1, 40 a file (1381)	1286	1287
families 2 to 8, sampled (2410)	2196	2199

No answer is wrong on either build. The 92 problems the two builds disagreed on, run again one build at a time, are 88 answered here and 2 on master, those two answered here too. Of the other four, x^3 e^(-3/2 i arctan(a x)) is answered in 43 s unbudgeted. The other three, 1/(e^(5/2 i arctan(a x)) x^k) for k from 2 to 4, master declines in 2 s, and here they ran past two minutes -- not in this rule but in the scaling of the variable, which simplified the integrand the radical's substitution leaves with a and i in it. With #1718, which is in master now, they are declined in 4 to 5 s on the merge below, where master 7a7eed7c declines them in about 1.

The suite passes on the commit measured, 0e882559, 14,564 tests, and so does the allocation gate. Master 7a7eed7c is merged in since, without conflicts, and the calculus tests pass on that merge, 3,966 of them.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura