\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx \]
Optimal antiderivative \[ \frac {11 a^{4} c^{4} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{5}}{48}-\frac {11 a^{5} c^{4} \left (1-\frac {1}{a x}\right )^{\frac {5}{2}} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{6}}{48}+\frac {11 a^{6} c^{4} \left (1-\frac {1}{a x}\right )^{\frac {7}{2}} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{7}}{56}-\frac {11 a^{7} c^{4} \left (1-\frac {1}{a x}\right )^{\frac {9}{2}} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{8}}{72}+\frac {a^{8} c^{4} \left (1-\frac {1}{a x}\right )^{\frac {11}{2}} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{9}}{9}+\frac {55 c^{4} \operatorname {arctanh}\! \left (\sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{128 a}+\frac {55 a \,c^{4} \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} x^{2} \sqrt {1-\frac {1}{a x}}}{384}+\frac {11 a^{2} c^{4} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}} x^{3} \sqrt {1-\frac {1}{a x}}}{192}-\frac {11 a^{3} c^{4} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{4} \sqrt {1-\frac {1}{a x}}}{64}+\frac {55 c^{4} x \sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}}{128} \]
command
Int[(c - a^2*c*x^2)^4/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
\[ \frac {1}{9} a^8 c^4 x^9 \left (1-\frac {1}{a x}\right )^{11/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{72} a^7 c^4 x^8 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {11}{56} a^6 c^4 x^7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{48} a^5 c^4 x^6 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {11}{48} a^4 c^4 x^5 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{64} a^3 c^4 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {11}{192} a^2 c^4 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {55 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{128 a}+\frac {55}{384} a c^4 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {55}{128} c^4 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \]