Integrate a quotient whose numerator is not constant (#233)

[Rafael-SOWNet]

Part of #233 (the partial-fractioning line), and a gap next to it.

What was missing

The table matched k/(ax^2 + bx + c) only when k did not mention x. Anything with a variable on top fell through every solver:

"x / (x ^ 2 + 2 * x + 5)".ToEntity().Integrate("x")      // integral(x / (x^2 + 2x + 5), x)
"(x + 3) / (x ^ 2 + 3 * x + 2)".ToEntity().Integrate("x")  // integral(...)
"x / (x + 1)".ToEntity().Integrate("x")                  // integral(...)

What this adds

A linear numerator over a quadratic. px + q is a multiple of the denominator's derivative plus a constant:

px + q = (p/2a)(2ax + b) + (q − pb/2a)

The first part integrates to ln|ax^2 + bx + c| and the second is the constant-numerator case that IntegrateRationalQuadratic already handles, including its split on the sign of the discriminant. So this is only the rewrite — no new closed forms and no new cases to get wrong.

A linear numerator over a linear denominator, which is the same rewrite one degree down: the quotient is the constant p/b plus a remainder over the divisor.

Both are guarded on the leading coefficient being a non-zero number, since the rewrite divides by it. A quadratic denominator whose quadratic term vanishes falls through to the linear arm rather than being divided by zero, and there is a test for that.

1/cos(u)^2 and 1/sin(u)^2. Written that way at least as often as sec(u)^2 and csc(u)^2, and none of the four shapes was recognised. sec and csc themselves already were.

Verification

• 21 new tests. Each answer is checked by differentiating it back and comparing against the integrand at points inside its domain, so a wrong closed form fails rather than an unexpected spelling of a right one.
• 3901 unit tests, 0 failed. 127 F# tests, 0 failed. Both target frameworks build.
• Solver corpus 88/117 -> 91/117, still 0 wrong, 0 error, 0 timeout. int:table and int:rational-quadratic reach 100%; int:partial-fractions goes from 2 out of 5 to 3.
• A property checker over 151 expressions (1320 checks) is clean.

Not covered

x^2/(x^4 + 1) and 1/(x^3 + 1) — the two remaining int:partial-fractions entries — still return unevaluated. Both need the denominator factored before the fractions can be split, and x^4 + 1 factors only over the irrationals. That is a separate piece of work.

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Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 80.43%. Comparing base (90c00a8) to head (c15cf90).
⚠️ Report is 56 commits behind head on master.
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Additional details and impacted files

@@            Coverage Diff             @@
##           master     #681      +/-   ##
==========================================
- Coverage   80.99%   80.43%   -0.57%     
==========================================
  Files         155      156       +1     
  Lines       13687    12934     -753     
  Branches     1957     2126     +169     
==========================================
- Hits        11086    10403     -683     
+ Misses       1990     1923      -67     
+ Partials      611      608       -3     

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[Rafael-SOWNet]

Rebased onto master now that #675 has landed. The two touch the same switch in IntegralPatterns.cs; the arms are disjoint and both are kept. Order was checked — TryGetPolyQuadratic and TryGetPolyLinear both fail on a root, so the general quotient arms added here cannot steal the radical cases #675 added above them. 4017 tests, none failing.