\[ \int \frac {e^{-\tanh ^{-1}(a x)}}{c-a c x} \, dx \]
Optimal antiderivative \[ \frac {\arcsin \! \left (a x \right )}{a c} \]
command
integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a*c*x+c),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2} x^{2} - 1}\, dx}{c} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}}{c} \]