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