\[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx \]
Optimal antiderivative \[ -\frac {a^{4} c^{3} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{5}}{6}+\frac {a^{5} c^{3} \left (1-\frac {1}{a x}\right )^{\frac {5}{2}} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{6}}{6}-\frac {a^{6} c^{3} \left (1-\frac {1}{a x}\right )^{\frac {7}{2}} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{7}}{7}-\frac {5 c^{3} \operatorname {arctanh}\! \left (\sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{16 a}-\frac {5 a \,c^{3} \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} x^{2} \sqrt {1-\frac {1}{a x}}}{48}-\frac {a^{2} c^{3} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}} x^{3} \sqrt {1-\frac {1}{a x}}}{24}+\frac {a^{3} c^{3} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{4} \sqrt {1-\frac {1}{a x}}}{8}-\frac {5 c^{3} x \sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}}{16} \]
command
Int[(c - a^2*c*x^2)^3/E^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
\[ -\frac {1}{7} a^6 c^3 x^7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {1}{6} a^5 c^3 x^6 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {1}{6} a^4 c^3 x^5 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {1}{8} a^3 c^3 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {1}{24} a^2 c^3 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {5 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{16 a}-\frac {5}{48} a c^3 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {5}{16} c^3 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \]