A root of the cosine beside a half-odd power of a + a sec is integrated wrongly where the cosine is negative #1581

sqrt(cos(c + d*x))/sqrt(a + a*sec(c + d*x)) is integrated wrongly where cos(c + d x) is negative.

Found by work/intbench on Rubi's 4.5.1.2 and 4.5.3.1, master a8a2b7e: seven problems come back wrong. All seven are a root of the cosine beside a half-odd power of a + a sec:

• sqrt(cos(c+d*x))/sqrt(a+a*sec(c+d*x)) (4.5.1.2:489)
• 1/(sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x))) (4.5.1.2:490)
• sqrt(cos(c+d*x))/(a+a*sec(c+d*x))^(3/2) (4.5.1.2:496)
• sqrt(cos(c+d*x))/(a+a*sec(c+d*x))^(5/2) (4.5.1.2:502)
• (A+B*sec(c+d*x))*sqrt(cos(c+d*x))/sqrt(a+a*sec(c+d*x)) (4.5.3.1:649)
• (A+B*sec(c+d*x))*sqrt(cos(c+d*x))/(a+a*sec(c+d*x))^(3/2) (4.5.3.1:655)
• (A+B*sec(c+d*x))*sqrt(cos(c+d*x))/(a+a*sec(c+d*x))^(5/2) (4.5.3.1:662)

Where the cosine is positive, all seven differentiate back (a = 1.3, c = 0.2, d = 1, A = 0.7, B = 1.1, at x = 0.3, 0.6, 0.9, 1.1). Where it is negative, the integrand is still real: sqrt(cos) and the root of a + a sec are both imaginary there. None of the seven differentiates back at x = 1.9, 2.2, 2.5, 2.9. By quadrature from 1.9 to 2.9, Simpson gives 3.6947 for the first, and master's answer gives F(2.9) - F(1.9) = -3.6947. The next three are also the exact negative of the quadrature there. The three with A + B sec(c + d x) differ from it by more than a sign.

The route, traced on master:

1. SolveByTakingAConstantOutOfAFractionalPower takes sqrt(a) out.
2. SolveBySubstitution takes u = c + d x.
3. SolveByTheSignOfTheComplement writes the integrand in w = cos(u).
4. SolveBySubstitution takes v = sqrt(w). For the first integrand, that leaves -2 v^2/(sqrt(1 + 1/v^2) sqrt(1 - v^4)).
5. SolveByCombiningRadicals writes the product of roots as sqrt(v^4 - v^8 + v^2 - v^6)/v^2.

Step 5 is right for a real v, where sqrt(v^4) = v^2. Wherever the cosine is negative, v = sqrt(w) is imaginary, sqrt(v^4) = -v^2, and the answer comes back negated.

SolveBySubstitution already conditions the answer on the radicand after an even root, provided w >= 0, for this reason. Its comment names (1 + sec(x))^(5/2) sqrt(cos(x)). It does so only when the answer below read a sign or a modulus of v (if (asTheRoot != resultInU)). Here the rules below wrote v^2 for sqrt(v^4) and read neither.

Tested: with that condition attached to every answer through an even root (a probe, not a proposal), all seven come back conditioned on cos(c + d x) >= 0: the first says provided a > 0 and cos(c + d * x) >= 0, where master says provided a > 0. They still differentiate back where the cosine is positive, and claim nothing where it is negative. The probe also conditions answers that hold on both sides, which a fix would have to avoid.