\[ \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {2 i \arctanh \left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {i b +a}}\right )}{d \sqrt {i b +a}} \]
command
integrate((1-I*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {4 \, \arctan \left (-\frac {2 \, {\left (-i \, \sqrt {b \tan \left (d x + c\right ) + a} a - i \, \sqrt {a^{2} + b^{2}} \sqrt {b \tan \left (d x + c\right ) + a}\right )}}{\sqrt {2 \, a + 2 \, \sqrt {a^{2} + b^{2}}} a + i \, \sqrt {2 \, a + 2 \, \sqrt {a^{2} + b^{2}}} b + \sqrt {a^{2} + b^{2}} \sqrt {2 \, a + 2 \, \sqrt {a^{2} + b^{2}}}}\right )}{\sqrt {2 \, a + 2 \, \sqrt {a^{2} + b^{2}}} d {\left (\frac {i \, b}{a + \sqrt {a^{2} + b^{2}}} + 1\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________