\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (6+n \right ) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}} \left (-a c x +c \right )^{1+\frac {n}{2}}}{a \left (n^{2}+6 n +8\right )}+\frac {2 \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}} x \left (-a c x +c \right )^{1+\frac {n}{2}}}{4+n} \]
command
integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(1+1/2*n),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {2 \, {\left (a^{2} \left (-c\right )^{\frac {1}{2} \, n} c {\left (n + 2\right )} x^{2} - 4 \, a \left (-c\right )^{\frac {1}{2} \, n} c x - \left (-c\right )^{\frac {1}{2} \, n} c {\left (n + 6\right )}\right )} {\left (a x + 1\right )}^{\frac {1}{2} \, n}}{{\left (n^{2} + 6 \, n + 8\right )} a} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int {\left (-a c x + c\right )}^{\frac {1}{2} \, n + 1} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]________________________________________________________________________________________