\[ \int \frac {1}{\left (c+a^2 c x^2\right ) \text {ArcTan}(a x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2}{a c \sqrt {\arctan \left (a x \right )}} \]
command
integrate(1/(a**2*c*x**2+c)/atan(a*x)**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {2}{a c \sqrt {\operatorname {atan}{\left (a x \right )}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \begin {cases} - \frac {2}{a c \sqrt {\operatorname {atan}{\left (a x \right )}}} & \text {for}\: c \neq 0 \\\tilde {\infty } \int \frac {1}{\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]________________________________________________________________________________________