\[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx \]
Optimal antiderivative \[ \frac {a^{4} c^{2} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{5}}{5}+\frac {3 c^{2} \operatorname {arctanh}\! \left (\sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{8 a}+\frac {a \,c^{2} \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} x^{2} \sqrt {1-\frac {1}{a x}}}{8}+\frac {a^{2} c^{2} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}} x^{3} \sqrt {1-\frac {1}{a x}}}{20}-\frac {3 a^{3} c^{2} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{4} \sqrt {1-\frac {1}{a x}}}{20}+\frac {3 c^{2} x \sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}}{8} \]
command
Int[E^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
\[ \frac {1}{5} a^4 c^2 x^5 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {3}{20} a^3 c^2 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {1}{20} a^2 c^2 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {3 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{8 a}+\frac {1}{8} a c^2 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {3}{8} c^2 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \]