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SolveBySecantPowerReduction(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

A whole power of the secant or cosecant, brought down two at a time by the standard
reduction until the power rule for the first or the zeroth takes over.

Remarks

1/cos(x)^2 was answered and 1/cos(x)^6 was not. The rule for powers of
sine and cosine reads a positive exponent, so a negative one — which is what a
secant or cosecant is — matched nothing above the square, and the square was answered
only because it is a standard integral in its own right.
This is Rubi's rule 4.5.1.1, ported:

Int[(b*csc[c+d*x])^n] := -b*Cos[c+d*x]*(b*Csc[c+d*x])^(n-1)/(d*(n-1))
                         + b^2*(n-2)/(n-1)*Int[(b*Csc[c+d*x])^(n-2)]
    /; GtQ[n,1] && IntegerQ[2*n]

which for the secant reads
∫sec^n = sec^(n-2) tan/(a(n-1)) + ((n-2)/(n-1)) ∫sec^(n-2), and for the
cosecant the same with -cot and the sign of the first term flipped. Each step
takes two off the exponent, so it ends at n = 1 — a standard integral here
already — or at n = 0, which is x.
Read through both spellings. A secant is sec(u), and it is also
cos(u)^(-n) and 1/cos(u)^n; all three arrive here, and answering one of
them and not the others is the defect this file has had five times over.
https://github.com/asc-community/AngouriMath/issues/718

AngouriMath

Table of contents

AngouriMath

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

SolveBySecantPowerReduction(AngouriMath.Entity,AngouriMath.Entity.Variable)