\[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx \]
Optimal antiderivative \[ -\frac {5 c \,\operatorname {arctanh}\! \left (\sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{2 a}-\frac {5 a c \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} x^{2} \sqrt {1-\frac {1}{a x}}}{6}-\frac {a^{2} c \left (1+\frac {1}{a x}\right )^{\frac {5}{2}} x^{3} \sqrt {1-\frac {1}{a x}}}{3}-\frac {5 c x \sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}}{2} \]
command
Int[E^(3*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
\[ -\frac {1}{3} a^2 c x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {5 c \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{2 a}-\frac {5}{6} a c x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {5}{2} c x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \]