\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx \]
Optimal antiderivative \[ \frac {8 c x \sqrt {1-\frac {1}{a^{2} x^{2}}}}{3 \sqrt {-a c x +c}}+\frac {2 x \sqrt {1-\frac {1}{a^{2} x^{2}}}\, \sqrt {-a c x +c}}{3} \]
command
integrate((-a*c*x+c)**(1/2)*((a*x-1)/(a*x+1))**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \int \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right )}\, dx \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {4 i c x \sqrt {\frac {1}{a c x + c}}}{3} + \frac {4 i c \sqrt {\frac {1}{a c x + c}}}{a} - \frac {2 i \left (- a c x + c\right )^{2} \sqrt {\frac {1}{a c x + c}}}{3 a c} \]