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SolveAProductOfPowersOfTrigonometricFunctionsThroughTheTangent​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable,​System.​Boolean)

 Method (no overloads)

Summary

A product of powers of the trigonometric functions of one argument, some of them not
whole, which is sin(z)^M cos(z)^N times a constant on every interval where both
are continuous: with M + N even integrated as
tan(z)^M (1 + tan(z)^2)^(-(M + N)/2), and with M or N an odd whole
number as the plain product, Chebyshev's other two cases.

Remarks

sqrt(c sin(x))/sqrt(d cos(x)) is sqrt(c/d) sqrt(tan(x)) wherever both
roots are real, and was declined, with the rest of such products the corpus has, a
constant inside a root among them: (d csc(x))^(3/2) sqrt(c sec(x)),
(a sin(x))^(5/2) sqrt(b sec(x)), (d sec(x))^(5/2) sqrt(b tan(x)). Each is
a power of the sine times a power of the cosine, and with the exponents adding up to an
even number it is a function of the tangent: sin^M cos^N is tan^M cos^(M + N),
and cos^(M + N) is (1 + tan^2)^(-(M + N)/2) exactly. Rubi's 4.1.0 to
4.6.0.
 The constant is not written: the answer is the integrand times the antiderivative of the
tangent's form over that form, a quotient whose logarithmic derivative is zero -- both
halves have (M cot(z) - N tan(z)) z' -- so it is constant wherever it is
continuous, and the answer holds on every interval where the integrand is, the constants
inside the roots whatever their signs.
https://github.com/asc-community/AngouriMath/issues/718

























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