\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}}}{a \,c^{2} \left (2+n \right )} \]
command
integrate(exp(n*acoth(a*x))/(-a*c*x+c)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {x e^{\frac {i \pi n}{2}}}{c^{2}} & \text {for}\: a = 0 \\\tilde {\infty } x e^{\infty n} & \text {for}\: a = \frac {1}{x} \\- \frac {a x \operatorname {acoth}{\left (a x \right )}}{a^{2} c^{2} x e^{2 \operatorname {acoth}{\left (a x \right )}} - a c^{2} e^{2 \operatorname {acoth}{\left (a x \right )}}} - \frac {\operatorname {acoth}{\left (a x \right )}}{a^{2} c^{2} x e^{2 \operatorname {acoth}{\left (a x \right )}} - a c^{2} e^{2 \operatorname {acoth}{\left (a x \right )}}} & \text {for}\: n = -2 \\- \frac {a x e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} c^{2} n x + 2 a^{2} c^{2} x - a c^{2} n - 2 a c^{2}} - \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} c^{2} n x + 2 a^{2} c^{2} x - a c^{2} n - 2 a c^{2}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \begin {cases} \text {NaN} & \text {for}\: a = \frac {1}{x} \wedge c = 0 \wedge n = -2 \\\tilde {\infty } x e^{\infty n} & \text {for}\: a = \frac {1}{x} \\\tilde {\infty } \int e^{n \operatorname {acoth}{\left (a x \right )}}\, dx & \text {for}\: c = 0 \\- \frac {a x \operatorname {acoth}{\left (a x \right )}}{a^{2} c^{2} x e^{2 \operatorname {acoth}{\left (a x \right )}} - a c^{2} e^{2 \operatorname {acoth}{\left (a x \right )}}} - \frac {\operatorname {acoth}{\left (a x \right )}}{a^{2} c^{2} x e^{2 \operatorname {acoth}{\left (a x \right )}} - a c^{2} e^{2 \operatorname {acoth}{\left (a x \right )}}} & \text {for}\: n = -2 \\- \frac {a x e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} c^{2} n x + 2 a^{2} c^{2} x - a c^{2} n - 2 a c^{2}} - \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} c^{2} n x + 2 a^{2} c^{2} x - a c^{2} n - 2 a c^{2}} & \text {otherwise} \end {cases} \]________________________________________________________________________________________