\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx \]
Optimal antiderivative \[ -\frac {16 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{-2+\frac {n}{2}} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\! \left (4,2-\frac {n}{2};3-\frac {n}{2};\frac {a -\frac {1}{x}}{a +\frac {1}{x}}\right )}{a \left (4-n \right )} \]
command
Int[E^(n*ArcCoth[a*x])*(c - a^2*c*x^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {16 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-4}{2}} \operatorname {Hypergeometric2F1}\left (4,2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \]