\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx \]
Optimal antiderivative \[ \frac {3 a^{3} c^{3} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}} x^{4}}{8}-\frac {3 a^{4} c^{3} \left (1-\frac {1}{a x}\right )^{\frac {5}{2}} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}} x^{5}}{10}+\frac {3 a^{5} c^{3} \left (1-\frac {1}{a x}\right )^{\frac {7}{2}} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}} x^{6}}{14}-\frac {a^{6} c^{3} \left (1-\frac {1}{a x}\right )^{\frac {9}{2}} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}} x^{7}}{7}+\frac {9 c^{3} \operatorname {arctanh}\! \left (\sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{16 a}+\frac {3 a \,c^{3} \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} x^{2} \sqrt {1-\frac {1}{a x}}}{16}-\frac {3 a^{2} c^{3} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}} x^{3} \sqrt {1-\frac {1}{a x}}}{8}+\frac {9 c^{3} x \sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}}{16} \]
command
Int[(c - a^2*c*x^2)^3/E^(3*ArcCoth[a*x]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {1}{7} a^6 c^3 x^7 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {3}{14} a^5 c^3 x^6 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {3}{10} a^4 c^3 x^5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {3}{8} a^3 c^3 x^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {3}{8} a^2 c^3 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {9 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{16 a}+\frac {3}{16} a c^3 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {9}{16} c^3 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \]