The third trigonometric substitution, and the sign that makes its second interval right

[Rafael-SOWNet]

sqrt(9*x^2 - 1)/x^2, 1/(x^3*sqrt(x^2 - 16)) and (x^2 - 10)^(5/2)/x had no
antiderivative. #1273 shipped the tangent and sine substitutions and left this one
out, saying its domain is two intervals rather than one and that the answer owed a
condition it did not write. That was the right call at the time and the condition
turns out not to be what it needs.

A > 0, c > 0   y = r tan(t)   ->  sin^m cos^(-m-k-2)     one interval, all of R
A > 0, c < 0   y = r sin(t)   ->  sin^m cos^(k+1)        one interval, between the roots
A < 0, c > 0   y = r sec(t)   ->  sin^(k+1) cos^(-m-k-2) two intervals, outside them

There is no fourth: both coefficients negative is a radicand negative everywhere,
which has no real integrand to integrate, and it is declined.

The sign, which took two attempts

(a tan(t)^2)^(k/2) is a^(k/2) |tan(t)|^k, and tan(t) is negative for t in
(pi/2, pi) -- which is where y < -r lands. Writing tan(t)^k there is wrong by
a sign for odd k.

The first attempt patched the three back-substitutions, putting an absolute value
on sin(t) and a signum on tan(t). That got y > r right and left eight of
thirteen shapes still sign-reversed on the other side, because the sign it is
wrong by depends on the power outside the radical as well and the three
replacements do not compose into it.

What is true without cases: the integrand y^m (b y^2 - a)^(k/2) is even or odd
under y -> -y according to m, so an antiderivative F valid for y > r gives
the whole answer as sign(y)^(m+1) F(|y|). So everything is written in |y|,
where the substitution's own interval is the principal one and every sign is
positive, and the reflection is applied once at the end. Thirteen of thirteen.

A whole power is not this substitution's, and that is correctness

(2*x + 3*x^2)^2 completes the square to a negative constant, so it reaches the
secant branch -- and it is a polynomial, real on the whole line, where the
substitution is defined only outside |y| > r. It came back with the sign reversed
at x = -0.5. That is a wrong answer, not a missing condition, and the branch now
takes only an odd half-power, where the integrand is a square root of something
negative in the middle and so is undefined exactly where the substitution is.

Caught by ExpandedPowerIntegralTest, which exists for that shape and fired.

Measured

Every case checked on both intervals -- two points above r and two below
-r. An answer right only to the right of the gap passes a one-sided test, which
is how the first attempt looked correct.

master (5070a363)   5/13 answered, 0 wrong
with it            13/13 answered, 0 wrong

Rubi corpus, 463-problem sample, both arms in the foreground:

master (5070a363)   255/463, 0 wrong, 3 timeouts, 158 s
with it             257/463, 0 wrong, 3 timeouts, 158 s

Two more, no timeout added, wall clock unmoved. The five test suites are green.

RootOverSquareIntegralsTest pinned 1/(x^3*sqrt(x^2 - 1)) as unevaluated and it
is now answered. That test is about its own formula rather than about the
integrand, so it now says so and checks the answer instead of its absence -- the
second case it pins is still unevaluated, since a negative power of the variable
beside a quadratic with a linear term is outside this rule too.

Part of #718.

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