\[ \int \frac {e^{-\text {ArcTan}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {24 \,{\mathrm e}^{-\arctan \left (a x \right )}}{85 a \,c^{3}}+\frac {\left (4 a x -1\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{17 a \,c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {12 \left (-2 a x +1\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{85 a \,c^{3} \left (a^{2} x^{2}+1\right )} \]
command
integrate(1/exp(atan(a*x))/(a**2*c*x**2+c)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {24 a^{4} x^{4}}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} + \frac {24 a^{3} x^{3}}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} - \frac {60 a^{2} x^{2}}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} + \frac {44 a x}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} - \frac {41}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} & \text {for}\: a \neq 0 \\\frac {x}{c^{3}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________