\[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx \]
Optimal antiderivative \[ \frac {5 a^{6} c^{4} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}} \left (1+\frac {1}{a x}\right )^{\frac {11}{2}} x^{7}}{72}-\frac {7 a^{7} c^{4} \left (1-\frac {1}{a x}\right )^{\frac {5}{2}} \left (1+\frac {1}{a x}\right )^{\frac {11}{2}} x^{8}}{72}+\frac {a^{8} c^{4} \left (1-\frac {1}{a x}\right )^{\frac {7}{2}} \left (1+\frac {1}{a x}\right )^{\frac {11}{2}} x^{9}}{9}+\frac {35 c^{4} \operatorname {arctanh}\! \left (\sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{128 a}+\frac {35 a \,c^{4} \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} x^{2} \sqrt {1-\frac {1}{a x}}}{384}+\frac {7 a^{2} c^{4} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}} x^{3} \sqrt {1-\frac {1}{a x}}}{192}+\frac {a^{3} c^{4} \left (1+\frac {1}{a x}\right )^{\frac {7}{2}} x^{4} \sqrt {1-\frac {1}{a x}}}{64}+\frac {a^{4} c^{4} \left (1+\frac {1}{a x}\right )^{\frac {9}{2}} x^{5} \sqrt {1-\frac {1}{a x}}}{144}-\frac {5 a^{5} c^{4} \left (1+\frac {1}{a x}\right )^{\frac {11}{2}} x^{6} \sqrt {1-\frac {1}{a x}}}{144}+\frac {35 c^{4} x \sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}}{128} \]
command
Int[E^ArcCoth[a*x]*(c - a^2*c*x^2)^4,x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {1}{9} a^8 c^4 x^9 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{11/2}-\frac {7}{72} a^7 c^4 x^8 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}+\frac {5}{72} a^6 c^4 x^7 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}-\frac {5}{144} a^5 c^4 x^6 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}+\frac {1}{144} a^4 c^4 x^5 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {1}{64} a^3 c^4 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{192} a^2 c^4 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {35 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{128 a}+\frac {35}{384} a c^4 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {35}{128} c^4 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \]