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WithARootOverTheCosineOrTheSine​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable)

 Method (no overloads)

Summary

(a cos(x) + b sin(x))^(m/2) cos(x)^(n/2), m and n odd, written as
(a + b tan(x))^(m/2) cos(x)^((m + n)/2), and the sine's the same way with
a cot(x) + b; null where the integrand is not that shape.

Remarks

A^(m/2) c^(n/2) is (sqrt(A)/sqrt(c))^m c^((m + n)/2) for principal roots,
and sqrt(A)/sqrt(c) is sqrt(A/c) where A and c have one
sign: both roots real, or both imaginary. Where the signs differ the product is
imaginary, so with no other root of x in the integrand to make it real again,
the two agree wherever the integrand is real. Rubi's 4.7.7
csc(x) sqrt(cos(x) + sin(x))/cos(x)^(3/2) is
sqrt(1 + tan(x)) csc(x) sec(x) so, and that is the tangent substitution's.
 Only a linear A: sqrt(c^3) and c^(3/2) differ in sign where
c is negative, so a cubic over the cube of the cosine is not the same quotient.
https://github.com/asc-community/AngouriMath/issues/718

























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