\[ \int \frac {e^{2 \text {ArcTan}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {3 \,{\mathrm e}^{2 \arctan \left (a x \right )}}{40 a \,c^{3}}+\frac {{\mathrm e}^{2 \arctan \left (a x \right )} \left (2 a x +1\right )}{10 a \,c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \,{\mathrm e}^{2 \arctan \left (a x \right )} \left (a x +1\right )}{20 a \,c^{3} \left (a^{2} x^{2}+1\right )} \]
command
integrate(exp(2*atan(a*x))/(a**2*c*x**2+c)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {3 a^{4} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {6 a^{3} x^{3} e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {12 a^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {14 a x e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {13 e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} & \text {for}\: a \neq 0 \\\frac {x}{c^{3}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \begin {cases} \frac {3 a^{4} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {6 a^{3} x^{3} e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {12 a^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {14 a x e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {13 e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} & \text {for}\: c \neq 0 \\\tilde {\infty } \int e^{2 \operatorname {atan}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]________________________________________________________________________________________