\[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} x \, dx \]
Optimal antiderivative \[ \frac {44 \left (1+\frac {1}{x}\right )^{\frac {3}{2}} \left (1-x \right )^{\frac {3}{2}}}{105 \left (1-\frac {1}{x}\right )^{\frac {3}{2}}}-\frac {22 \left (1+\frac {1}{x}\right )^{\frac {3}{2}} \left (1-x \right )^{\frac {3}{2}} x}{35 \left (1-\frac {1}{x}\right )^{\frac {3}{2}}}+\frac {2 \left (1+\frac {1}{x}\right )^{\frac {3}{2}} \left (1-x \right )^{\frac {3}{2}} x^{2}}{7 \left (1-\frac {1}{x}\right )^{\frac {3}{2}}} \]
command
integrate(1/((-1+x)/(1+x))**(1/2)*(1-x)**(3/2)*x,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {2 \left (1 - x\right )^{\frac {7}{2}}}{7 \sqrt {- \frac {x}{- x - 1} + \frac {1}{- x - 1}}} - \frac {18 \left (1 - x\right )^{\frac {5}{2}}}{35 \sqrt {- \frac {x}{- x - 1} + \frac {1}{- x - 1}}} - \frac {4 \left (1 - x\right )^{\frac {3}{2}}}{105 \sqrt {- \frac {x}{- x - 1} + \frac {1}{- x - 1}}} - \frac {16 \sqrt {1 - x}}{105 \sqrt {- \frac {x}{- x - 1} + \frac {1}{- x - 1}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________