AngouriMath
SolveASumOfPowersOfAnInverseOrErrorFunctionByPartsDownThePowers(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)
Method (no overloads)
Summary
A sum of terms each a power of one function K whose derivative is free of it -- an
inverse trigonometric function or an error function -- times a cofactor free of it,
integrated by parts down the powers:∫ A K^p = F K^p - ∫ p F K^(p - 1) K' for
F = ∫ A , the cofactor of the top power, and the last integral joins the terms a
power below. Where every power cancels so, the parts are the antiderivative:
x^3/((1 + a^2 x^2) atan(a x)^3) - 3 x^2/(2 a atan(a x)^2) is
(-x^3/(2 a atan(a x)^2))' , in one step, and
x asin(x)^(3/2)/(1 - x^2)^2 - 3 x/(8 (1 - x^2) sqrt(asin(x))) in two. Where what is
left is free ofK , it is asked: erf(b x) e^(-b^2 x^2) (1/x^3 + b^2/x) leaves
b e^(-2 b^2 x^2)/(sqrt(pi) x^2) .
inverse trigonometric function or an error function -- times a cofactor free of it,
integrated by parts down the powers:
power below. Where every power cancels so, the parts are the antiderivative:
left is free of
Remarks
at a time. Rubi's 5.1.4, 5.3.4 and 8.1 write such sums to test that the parts meet.
of it, the powers a whole number apart, at most five apart, and each step's
answer is checked by differentiating it back. A remainder free of
only for a top power of two terms or more with a cofactor among them that is not
algebraic, which is where the cofactors integrate together and not apart: a sum of
algebraic cofactors,
and this rule would only respell it. At the question asked only, and not inside its
own questions, which are free of
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