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