AngouriMath
SolveByEulerSubstitution(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)
Method (no overloads)
Summary
A rational function of x and one square root of a quadratic in x ,
rationalised by an Euler substitution and handed to the rational integrator.
rationalised by an Euler substitution and handed to the rational integrator.
Remarks
nothing wider;
integrands that other rules reduce to a rational function of
root -- the nested radicals of Bondarenko's suite under
by-parts remainders of Charlwood's inverse functions.
a > 0 :sqrt(Q) = t - sqrt(a) x , sox = (t^2 - c)/(2 sqrt(a) t + b) .c > 0 :sqrt(Q) = x t + sqrt(c) , sox = (2 sqrt(c) t - b)/(a - t^2) .c = 0 :sqrt(Q) = x t , sox = b/(t^2 - a) -- the third substitution at
the root the quadratic has at zero.
integrand becomes a rational function of
a rational number is taken first. A root of a symbol is not taken, except through the
sum the integrand writes, below. The result goes to the rational integrator directly --
long division, the splits, the Hermite reduction, the binomial rule -- and not back
into the chain: an earlier Euler rule rewrote and handed on, and its cost was in the
open search below it and in a
applied at all (https://github.com/asc-community/AngouriMath/issues/1265). This one
decides by reading the tree, in microseconds, and finishes in one closed step or not
at all.
splits cannot factor is declined by them at a cost that grows with it. Asked, not
volunteered, like every rule that lands on a search of any size.
variable, and its powers are powers of it: Rubi's
answers through such a sum or not at all, which is how it is asked before the split
that would expand the powers of
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