\[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d +c}}\right )}{\left (i a +b \right ) \left (-i d +c \right )^{\frac {3}{2}} f}-\frac {\arctanh \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d +c}}\right )}{\left (i a -b \right ) \left (i d +c \right )^{\frac {3}{2}} f}-\frac {2 b^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {b}\, \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-a d +b c}}\right )}{\left (a^{2}+b^{2}\right ) \left (-a d +b c \right )^{\frac {3}{2}} f}+\frac {2 d^{2}}{\left (-a d +b c \right ) \left (c^{2}+d^{2}\right ) f \sqrt {c +d \tan \left (f x +e \right )}} \]
command
integrate(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]