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

An exponential of a multiple of arctan(a x) times a quadratic polynomial times a
power of c + a^2 c x^2, where it is an exact derivative:
(e^(N arctan(a x)) (alpha + beta x) Q^s)' is
c e^(N arctan(a x)) Q^(s - 1) (a N alpha + beta + (a N beta + 2 s a^2 alpha) x + a^2 beta (1 + 2 s) x^2) for Q = c + a^2 c x^2, since Q^s/(1 + a^2 x^2) = c Q^(s - 1). The
constant and squared coefficients give alpha and beta; the linear one is
the condition. Rubi's 5.3.6:504, e^(i n arctan(a x)) x^2 (c + a^2 c x^2)^(-1 - n^2/2),
meets it exactly, and was left unevaluated: as powers of the two linears its exponents
do not sum to a whole number.
https://github.com/asc-community/AngouriMath/issues/718