Add five families of standard integrals (#233)

[Rafael-SOWNet]

Towards #233. Three categories
of my integral corpus sat at 0%, and none of them needed new machinery — each is a table
entry that was simply missing.

k / sqrt(a x² + b x + c)

Nothing integrated 1 / sqrt(1 - x^2) at all. It is an arcsine where a < 0 and a
logarithm where a > 0, returned as a piecewise on that sign, the way the rational
quadratic beside it in the table already is.

∫ 1/√(1 − x²) dx     ∫ 2/√(9 − x²) dx     ∫ 1/√(x² + 1) dx
∫ 1/√(x² − 1) dx     ∫ 1/√(2x + 3) dx  ← the a = 0 branch

sqrt(a x² + b x + c)

sqrt(1 - x^2) and sqrt(x^2 + 1) had no antiderivative either. Integrating by parts
once leaves the reciprocal form above, so it is written in terms of that rather than
worked out again. Only where the leading coefficient is a number other than zero — with
a = 0 this is the square root of something linear, which the power rule already
integrates, and there is a test that it is left alone.

An exponential times a sine or cosine

Integrating by parts twice returns the integral it started from, so the general solver
cycles rather than terminating. Solving that equation for the integral once gives a
closed form. The rate is read off B^(px + q) as p · ln(B), so any base and any linear
argument work — 2^x · cos(x) and e^(2x) · sin(3x) as much as e^x · sin(x).

The inverse trigonometric functions

Each is integration by parts against 1. The by-parts solver looks for a product to
split, so it never saw them.

∫ arcsin(x) dx   ∫ arccos(x) dx   ∫ arctan(x) dx   ∫ arccotan(x) dx

Any whole power of sine times any whole power of cosine

sin(x)^3 and sin(x)^2 * cos(x)^2 had no antiderivative, and sin(x)^2 and
cos(x)^2 each had one written out by hand. There are two cases and this does both:

• An odd power gives a factor to peel off as the differential —
  sin^(2k+1) = sin · (1 - cos²)^k — leaving a polynomial in the other function, so the
  integral is a finite sum.
• Both even, and there is no such factor, so the halved-angle identities go in and
  the result is integrated again. The total power halves each time, so it ends.

That subsumes the two hand-written squares, which are gone. Where both powers are odd
either substitution serves and the answers differ by a constant; the sine one is tried
first, because sin(x)cos(x) is written sin(x)^2/2 rather than the equally correct
-cos(x)^2/2, and two existing tests say so.

A skipped test that is no longer skipped

TestIntegrationByPartsNonPolynomial carried Skip = "TODO: integration by parts multiple times". Five of its six cases pass now — both exponential-times-wave ones and all
three inverse trigonometric ones — so the theory runs. The sixth, ln|x|^3, wants by
parts three times over; it is split into its own skipped test rather than holding the
other five back.

I kept that test's arctan expectation working by writing the answer with ln(abs(...))
rather than ln(...). The argument 1 + x² is positive so the abs is redundant, but it
matches both that expectation and the convention the rest of the table already uses.

Result

On the 117-problem corpus: 82 → 92 solved. Three categories go to 100% --
int:arcsin-form and int:by-parts-cyclic from 0%, int:trig-powers from 50% --
and int:trig-sub from 0% to 67%. No integral category is left at 0%. What remains
in trig-sub is 1/(x^2 sqrt(x^2 - 1)), which wants a substitution rather than a table
entry.

Testing

UnitTests 3889 passed / 0 failed, FSharpWrapperUnitTests 127 passed / 0 failed, both
netstandard2.0 and net7.0 build.

43 new tests. Every one checks by differentiating the answer back and comparing at
points, rather than against a printed form — and at points inside each integrand's own
domain: 1/sqrt(x^2 - 1) is imaginary on (-1, 1), so testing it there would say
nothing about whether the answer is right. The last group asserts that the neighbouring
table entries are unaffected.

🤖 Generated with Claude Code

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Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 80.32%. Comparing base (90c00a8) to head (0be8877).
⚠️ Report is 51 commits behind head on master.
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Additional details and impacted files

@@            Coverage Diff             @@
##           master     #675      +/-   ##
==========================================
- Coverage   80.99%   80.32%   -0.68%     
==========================================
  Files         155      156       +1     
  Lines       13687    12780     -907     
  Branches     1957     2088     +131     
==========================================
- Hits        11086    10265     -821     
+ Misses       1990     1916      -74     
+ Partials      611      599      -12     

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