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SolveASymbolicPowerOfATrigonometricFunctionAsTheDerivativeOfAProduct(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

f^m (alpha + beta f^2) for a symbolic m, or a number that is not whole, and f a
sine, cosine, secant or cosecant of a linear u, where it is the derivative of a product: with u' = 1,
(cos sin^(m + 1))' = sin^m ((m + 1) - (m + 2) sin^2),
(sin cos^(m + 1))' = cos^m ((m + 2) cos^2 - (m + 1)),
(tan sec^m)' = sec^m ((m + 1) sec^2 - m) and (cot csc^m)' = -csc^m ((m + 1) csc^2 - m).

Remarks

The integrand is read as f^m times a polynomial in f with only the constant and
square terms, alpha + beta f^2, which matches p0 + p2 f^2 above where
beta p0 = alpha p2; the answer is (alpha/p0) G/u'. Rubi's 4.1.4.1, 4.2.4.1 and 4.5.4.1
sec(c + d x)^m (m - (1 + m) sec(c + d x)^2) and the like were declined.
The power is one of the integrand's factors, and one under the bar is f^(-m): only then is the
integrand f^m times the rest, and a power in one term of a sum, as in
-m sec^m + (1 + m) sec^2, is not a factor of it.
https://github.com/asc-community/AngouriMath/issues/718

AngouriMath

Table of contents

AngouriMath

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

SolveASymbolicPowerOfATrigonometricFunctionAsTheDerivativeOfAProduct(AngouriMath.Entity,AngouriMath.Entity.Variable)