AngouriMath

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WherePowersOfConjugateSumsAddToANegativeWhole(AngouriMath.Entity,AngouriMath.Entity.Variable,AngouriMath.Entity,System.ValueTuple{AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,System.Boolean},System.ValueTuple{AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,System.Boolean})

Method (no overloads)

Summary

sec(y) (a ± a sec(y))^m (c ∓ c sec(y))^n with m + n a negative whole
number -k and neither half-odd: with u = a ± a sec(y), the antiderivative is
the integrand over u' times the polynomial
(2a)^(-k) Σ_j C(k - 1, j) u^(j + 1) (2a - u)^(k - j)/(m + j + 1/2), for any m.

Remarks

Rubi's sec(e + f x) (a + a sec(e + f x))^m (c - c sec(e + f x))^(-1 - m) and its
-2 - m and -3 - m, 4.5.2.3, were declined. In u the integrand is
u^(m - 1/2) (2 - u/a)^(n - 1/2) up to a constant, which is v^(m - 1/2) (1 + v)^(k - 1) in v = u/(2a - u), a finite sum; written back, the powers of u and
2a - u the integrand has divide out and leave the polynomial. For k = 1 that is
-tan(y) (a + a sec(y))^m (c - c sec(y))^(-1 - m)/(2m + 1). Kept only where it
differentiates back to the integrand at the sampled points.
https://github.com/asc-community/AngouriMath/issues/718

AngouriMath

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