\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx \]
Optimal antiderivative \[ \frac {64 c^{2} \left (1-\frac {1}{a x}\right )^{3-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{-3+\frac {n}{2}} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\! \left (6,3-\frac {n}{2};4-\frac {n}{2};\frac {a -\frac {1}{x}}{a +\frac {1}{x}}\right )}{a \left (6-n \right )} \]
command
Int[E^(n*ArcCoth[a*x])*(c - a^2*c*x^2)^2,x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {64 c^2 \left (1-\frac {1}{a x}\right )^{3-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-6}{2}} \operatorname {Hypergeometric2F1}\left (6,3-\frac {n}{2},4-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (6-n)} \]