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SolveByPartsAgainstTheReciprocalOfASquaredBase(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

A quotient by the square of a base, by parts against the base's reciprocal, where the
base's derivative makes the rest exact: N/(M v^2) is g v'/v^2 with
g = N/(M v'), and v'/v^2 is the derivative of -1/v, so the integral
is -g/v plus the integral of g'/v; taken only where v cancels from
that, with the functions of x in it taken for indeterminates.

Remarks

x^2/(a x cos(a x) - sin(a x))^2 was declined, with the rest of Rubi's 4.7.7 by the
square of a x cos(a x) - sin(a x) or cos(a x) + a x sin(a x): the base's
derivative is -a^2 x sin(a x), so g is -x csc(a x)/a^2, and
g' is a^2 v/(a^2 sin(a x))^2, whose integral is -cot(a x)/a^3. Only
for a base with a function of x in it that is not a polynomial, since
a quotient of polynomials is the partial fractions', and only where v cancels,
since otherwise the remainder is the harder question.
https://github.com/asc-community/AngouriMath/issues/718

AngouriMath

Table of contents

AngouriMath

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

SolveByPartsAgainstTheReciprocalOfASquaredBase(AngouriMath.Entity,AngouriMath.Entity.Variable)