SolveByDistributingWholePowersOfQuotients​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable,​System.​Boolean)

A whole power of a quotient with a symbol in it, written as the quotient of the powers:
(c/(a + c u^2))^2 is c^2/(a + c u^2)^2, exact for a whole power. That is how
the substitution u = x^2 writes x/(a + c x^4)^2, having simplified
1/(a/c + u^2)^2, and the table reads the quotient of the powers and not the power
of the quotient: c^2/(a + c x^2)^2 was answered in a few milliseconds where
(c/(a + c x^2))^2 went to parts and was declined after twenty-six seconds, and
x/(a + c x^4)^2 was answered after forty, by another route. With numbers only it
is answered as it is already. Only a quotient of two polynomials in x:
the hyperbolic functions arrive as quotients of exponentials, and the imaginary tangent's
sum as A e^(i z)/cos(z), and written apart those went to slower routes --
cosh(c + d x)/(a + b tanh(c + d x)^2) from 17 ms to past five seconds. And only with
a monomial left above the bar, the table's x^m/(a + b x^n)^p: a sum there is
multiplied out by the rational rules, and the folded tanh(a + 2 ln(x))^2,
((e^(2a) x^4 - 1)/(e^(2a) x^4 + 1))^2, went from five seconds to forty-five that way.
Asked as the same question.
https://github.com/asc-community/AngouriMath/issues/718