\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx \]
Optimal antiderivative \[ -\frac {256 c^{3} \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{-4+\frac {n}{2}} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\! \left (8,4-\frac {n}{2};5-\frac {n}{2};\frac {a -\frac {1}{x}}{a +\frac {1}{x}}\right )}{a \left (8-n \right )} \]
command
Int[E^(n*ArcCoth[a*x])*(c - a^2*c*x^2)^3,x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {256 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-8}{2}} \operatorname {Hypergeometric2F1}\left (8,4-\frac {n}{2},5-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)} \]