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SolveByRothsteinTrager(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

Reads expr as a numerator over a denominator, in either of the two
spellings a quotient has here.

Remarks

1/(1 + x^3) is a Divf and (1 + x^3)^(-1) is a Powf, and
they are the same integrand. SolveByPartialFractions(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean) matched the first
and declined the second, so which of them was asked decided whether the integral came
back — and the second is not an exotic way to write it: it is what
SolveAsPolynomialTerm(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean) builds whenever it takes a factor out of a
quotient. That is why a/(1 + x^3) had no antiderivative while
1/(1 + x^3) and a * (1/(1 + x^3)) both did.
And the third spelling, a product with a negative power in it.3 u (4 - u^3)^(-1) is what Simplify makes of 3u/(4 - u^3), and it
is neither a Divf nor a bare Powf, so it was declined here -- and then
taken by integration by parts with v' = (4 - u^3)^(-1), which integrates to a
hundred-node sum of logarithms and arctangents that the search then spent thirty
seconds failing to integrate against 3u. Read as the quotient it is, the same
integrand is answered in a fifth of a second. The factors are gathered and one quotient
rebuilt from them, which is what makes the verdict independent of how the product
happens to be associated.
Only a negative whole power. A positive one is not a quotient; a fractional or
symbolic exponent is not one either, and (a/b)^(1/2) is not sqrt(a)/sqrt(b) on the branch cut. Nothing here rewrites a quotient back into a power, so this cannot
re-enter the mutual recursion Normalized(AngouriMath.Entity,AngouriMath.Entity.Variable) records.
https://github.com/asc-community/AngouriMath/issues/718

Summary

A rational function with rational coefficients whose denominator the splits above
could not take apart -- irreducible over the rationals past degree two, or with
real factors that carry the roots of something worse -- integrated by the Hermite
reduction and the Rothstein–Trager resultant, in real terms. See
RothsteinTrager. After SolveByPartialFractions(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean), so that
everything that answers keeps the form it gives.

AngouriMath

Table of contents

AngouriMath

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

SolveByRothsteinTrager(AngouriMath.Entity,AngouriMath.Entity.Variable)