An exponential times a trigonometric at the same frequency no longer divides by zero, and a phase is expanded

[Rafael-SOWNet]

x e^(-i x) cos(x) was answered NaN at every point of its domain — a wrong answer, not a decline.

The closed form int e^(ax)(c cos(bx) + d sin(bx)) = e^(ax)(…)/(a² + b²) is guarded against the resonance a = ±i b, where the product is a sum of two exponentials and the denominator vanishes. Both places that use it asked whether a² + b² evaluates to a number — and with a symbolic frequency it does not: (-i f)² + f² is -f² + f², which InnerSimplified does not collect. So the guard passed, the formula divided by a zero it could not see, and the answer was NaN. Both now test the resonance symbolically (IsZeroOnceSimplified, which sat directly above one of them).

On that footing, two capabilities:

• A phase is expanded. cos(f x + p) becomes cos(p) cos(f x) - sin(p) sin(f x) where an exponential stands beside it, because the closed rule reads one frequency and no phase — an exponential's own offset is a constant factor, a trigonometric's is not, and the reader said so in a comment ("a phase would want the angle-sum identity first"). Only beside an exponential: the expansion doubles every sine and cosine it touches, and sin(a + b x) alone is better answered by the closed rule that takes it whole.
• A + i A tan(z) below the bar is an exponential. A e^(i z)/cos(z), and A + i A cot(z) is i A e^(-i z)/sin(z) — exactly, since cos z + i sin z is e^(i z). Beside a polynomial that is the shape above. Below the bar only: above it the tangent's own rules answer in the tangent and more shortly, and before that restriction (a + i a tan(c + d x))^3 (A + B tan(c + d x))/tan(c + d x)^(5/2) went from 1.6 s to 4.2 s.

Measured (final build):

• Family 4 (familycap.sh 4 6): 284 → 286/422, 0 wrong, and unevaluated 93 → 85 — eight rows that produced nothing now produce answers (several graded unverifiable on the reals, since the integrands carry i and the harness samples the real line; each was verified separately as a complex function).
• Family 6: 377/417, 0 wrong, unchanged. 1774-problem suite: 1707, 0 wrong, 0 timeout, unchanged.
• Unit suite: 12,641 tests, 0 failed. bench.sh: PASSED on all 19 gated benchmarks.
• Twelve integrands verified by differentiating the answer and comparing against the integrand as complex numbers at five points — worst relative deviation 0 on every one, including x e^(2x) cos(3x + 1) and x² e^x sin(x + 2), which the phase expansion is for.

BREAKING-CHANGES.md carries the entry with the 2.5.0 column measured on a 2.5.0 build, and names the NaN as what master answered between the releases.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura