\[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (a^{3} \left (c^{2} C -2 B c d -C \,d^{2}-A \left (c^{2}-d^{2}\right )\right )-3 a \,b^{2} \left (c^{2} C -2 B c d -C \,d^{2}-A \left (c^{2}-d^{2}\right )\right )-3 a^{2} b \left (2 c \left (A -C \right ) d -B \left (c^{2}-d^{2}\right )\right )+b^{3} \left (2 c \left (A -C \right ) d -B \left (c^{2}-d^{2}\right )\right )\right ) x}{\left (c^{2}+d^{2}\right )^{2}}+\frac {\left (3 a^{2} b \left (c^{2} C -2 B c d -C \,d^{2}-A \left (c^{2}-d^{2}\right )\right )-b^{3} \left (c^{2} C -2 B c d -C \,d^{2}-A \left (c^{2}-d^{2}\right )\right )+a^{3} \left (2 c \left (A -C \right ) d -B \left (c^{2}-d^{2}\right )\right )-3 a \,b^{2} \left (2 c \left (A -C \right ) d -B \left (c^{2}-d^{2}\right )\right )\right ) \ln \left (\cos \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{2} f}+\frac {\left (-a d +b c \right )^{2} \left (b \left (3 c^{4} C -2 B \,c^{3} d +c^{2} \left (A +5 C \right ) d^{2}-4 B c \,d^{3}+3 A \,d^{4}\right )+a \,d^{2} \left (2 c \left (A -C \right ) d -B \left (c^{2}-d^{2}\right )\right )\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{d^{4} \left (c^{2}+d^{2}\right )^{2} f}+\frac {b^{2} \left (a d \left (3 c^{2} C -B c d +\left (A +2 C \right ) d^{2}\right )-b \left (3 c^{3} C -2 B \,c^{2} d +c \left (A +2 C \right ) d^{2}-B \,d^{3}\right )\right ) \tan \left (f x +e \right )}{d^{3} \left (c^{2}+d^{2}\right ) f}+\frac {b \left (3 c^{2} C -2 B c d +\left (2 A +C \right ) d^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )^{2}}{2 d^{2} \left (c^{2}+d^{2}\right ) f}-\frac {\left (A \,d^{2}-B c d +c^{2} C \right ) \left (a +b \tan \left (f x +e \right )\right )^{3}}{d \left (c^{2}+d^{2}\right ) f \left (c +d \tan \left (f x +e \right )\right )} \]
command
integrate((a+b*tan(f*x+e))**3*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(c+d*tan(f*x+e))**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________