\[ \int \frac {e^{-2 \text {ArcTan}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {{\mathrm e}^{-2 \arctan \left (a x \right )}}{8 a \,c^{2}}+\frac {\left (a x -1\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{4 a \,c^{2} \left (a^{2} x^{2}+1\right )} \]
command
integrate(1/exp(2*atan(a*x))/(a**2*c*x**2+c)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {a^{2} x^{2}}{8 a^{3} c^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 8 a c^{2} e^{2 \operatorname {atan}{\left (a x \right )}}} + \frac {2 a x}{8 a^{3} c^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 8 a c^{2} e^{2 \operatorname {atan}{\left (a x \right )}}} - \frac {3}{8 a^{3} c^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 8 a c^{2} e^{2 \operatorname {atan}{\left (a x \right )}}} & \text {for}\: a \neq 0 \\\frac {x}{c^{2}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \begin {cases} - \frac {a^{2} x^{2}}{8 a^{3} c^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 8 a c^{2} e^{2 \operatorname {atan}{\left (a x \right )}}} + \frac {2 a x}{8 a^{3} c^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 8 a c^{2} e^{2 \operatorname {atan}{\left (a x \right )}}} - \frac {3}{8 a^{3} c^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 8 a c^{2} e^{2 \operatorname {atan}{\left (a x \right )}}} & \text {for}\: c \neq 0 \\\tilde {\infty } \int e^{- 2 \operatorname {atan}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]________________________________________________________________________________________