\[ \int \frac {e^{\text {ArcTan}(a x)}}{\left (c+a^2 c x^2\right )^4} \, dx \]
Optimal antiderivative \[ \frac {144 \,{\mathrm e}^{\arctan \left (a x \right )}}{629 a \,c^{4}}+\frac {{\mathrm e}^{\arctan \left (a x \right )} \left (6 a x +1\right )}{37 a \,c^{4} \left (a^{2} x^{2}+1\right )^{3}}+\frac {30 \,{\mathrm e}^{\arctan \left (a x \right )} \left (4 a x +1\right )}{629 a \,c^{4} \left (a^{2} x^{2}+1\right )^{2}}+\frac {72 \,{\mathrm e}^{\arctan \left (a x \right )} \left (2 a x +1\right )}{629 a \,c^{4} \left (a^{2} x^{2}+1\right )} \]
command
integrate(exp(atan(a*x))/(a**2*c*x**2+c)**4,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {144 a^{6} x^{6} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {144 a^{5} x^{5} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {504 a^{4} x^{4} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {408 a^{3} x^{3} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {606 a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {366 a x e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {263 e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} & \text {for}\: a \neq 0 \\\frac {x}{c^{4}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \begin {cases} \frac {144 a^{6} x^{6} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {144 a^{5} x^{5} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {504 a^{4} x^{4} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {408 a^{3} x^{3} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {606 a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {366 a x e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {263 e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} & \text {for}\: c \neq 0 \\\tilde {\infty } \int e^{\operatorname {atan}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]________________________________________________________________________________________