\[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {5 \left (1-\frac {1}{x}\right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {2}\, \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right ) \sqrt {2}}{2 \left (1-x \right )^{\frac {3}{2}} \left (\frac {1}{x}\right )^{\frac {3}{2}}}-\frac {\left (1+\frac {1}{x}\right )^{\frac {3}{2}} x^{2} \sqrt {1-\frac {1}{x}}}{2 \left (1-x \right )^{\frac {3}{2}}}+\frac {5 \left (1-\frac {1}{x}\right )^{\frac {3}{2}} x^{2} \sqrt {1+\frac {1}{x}}}{2 \left (1-x \right )^{\frac {3}{2}}} \]
command
integrate(1/((-1+x)/(1+x))**(1/2)*x/(1-x)**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ 2 \left (\begin {cases} \sqrt {2} \left (\frac {\sqrt {2} \sqrt {- x - 1}}{2} - \operatorname {acos}{\left (\frac {\sqrt {2}}{\sqrt {1 - x}} \right )}\right ) & \text {for}\: \sqrt {1 - x} > - \sqrt {2} \wedge \sqrt {1 - x} < \sqrt {2} \end {cases}\right ) - 2 \left (\begin {cases} \frac {\sqrt {2} \left (\frac {\operatorname {acos}{\left (\frac {\sqrt {2}}{\sqrt {1 - x}} \right )}}{2} - \frac {\sqrt {2} \sqrt {1 - \frac {2}{1 - x}}}{2 \sqrt {1 - x}}\right )}{2} & \text {for}\: \sqrt {1 - x} > - \sqrt {2} \wedge \sqrt {1 - x} < \sqrt {2} \end {cases}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \int \frac {x}{\sqrt {\frac {x - 1}{x + 1}} \left (1 - x\right )^{\frac {3}{2}}}\, dx \]________________________________________________________________________________________