\[ \int \frac {1}{\sqrt {b \tan ^3(c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {2 \tan \left (d x +c \right )}{d \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}}-\frac {\arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) \sqrt {2}}{2 d \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}}-\frac {\arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) \sqrt {2}}{2 d \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}}-\frac {\ln \left (1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) \sqrt {2}}{4 d \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}}+\frac {\ln \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) \sqrt {2}}{4 d \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}} \]
command
integrate(1/(b*tan(d*x+c)^3)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{4} \, b^{2} {\left (\frac {2 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{4} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} + \frac {2 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{4} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} - \frac {\sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{4} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} + \frac {\sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{4} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} + \frac {8}{\sqrt {b \tan \left (d x + c\right )} b^{2} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )}\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________