\[ \int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (i a +b \right ) \arctanh \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d +c}}\right )}{\left (-i d +c \right )^{\frac {3}{2}} f}+\frac {\left (i a -b \right ) \arctanh \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d +c}}\right )}{\left (i d +c \right )^{\frac {3}{2}} f}+\frac {-2 a d +2 b c}{\left (c^{2}+d^{2}\right ) f \sqrt {c +d \tan \left (f x +e \right )}} \]
command
integrate((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 \[ \text {Timed out} \]