AngouriMath
SolveAPowerOfXTimesAnArctangentAndTheLogarithmOfItsQuadratic(AngouriMath.Entity,AngouriMath.Entity.Variable)
Method (no overloads)
Summary
A whole power of x times F = a + b atan(c x) times G = d + e ln(1 + c^2 x^2) ,
by parts on the power: withq = c^2 and n = k + 1 , the remainder is
(1/n) int x^n (b c G + 2 e q x F)/(1 + q x^2) , and x^n/(1 + q x^2) is a
polynomial plus(-1/q)^m x^(n - 2m)/(1 + q x^2) , m = floor(n/2) .
by parts on the power: with
polynomial plus
Remarks
For an odd k the two remainders over the quadratic, G/(1 + q x^2) and
x F/(1 + q x^2) , have no elementary antiderivative apiece and come with the same
factor, and together they are(F G)'/(b c) -- F' = b c/(1 + q x^2) ,
G' = 2 e q x/(1 + q x^2) -- so their integral is that factor times F G .
Integration by parts as the rules after it apply it never pairs them, and Rubi's 5.3.4
x^k (a + b atan(c x))(d + e ln(1 + c^2 x^2)) , k = 1 to 4 , timed out.
https://github.com/asc-community/AngouriMath/issues/718
factor, and together they are
Integration by parts as the rules after it apply it never pairs them, and Rubi's 5.3.4
https://github.com/asc-community/AngouriMath/issues/718
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