\[ \int \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}} \, dx \]
Optimal antiderivative \[ -\frac {\cot \! \left (d x +c \right ) \ln \! \left (\cos \! \left (d x +c \right )\right ) \left (b \left (\tan ^{p}\left (d x +c \right )\right )\right )^{\frac {1}{p}}}{d} \]
command
integrate((b*tan(d*x+c)^p)^(1/p),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {4 \, \pi {\left | b \right |}^{\left (\frac {1}{p}\right )} \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \tan \left (\frac {\pi \mathrm {sgn}\left (b\right )}{4 \, p} - \frac {\pi }{4 \, p}\right ) + {\left | b \right |}^{\left (\frac {1}{p}\right )} \log \left (\frac {4}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (\frac {\pi \mathrm {sgn}\left (b\right )}{4 \, p} - \frac {\pi }{4 \, p}\right )^{2} - 4 \, c {\left | b \right |}^{\left (\frac {1}{p}\right )} \tan \left (\frac {\pi \mathrm {sgn}\left (b\right )}{4 \, p} - \frac {\pi }{4 \, p}\right ) - {\left | b \right |}^{\left (\frac {1}{p}\right )} \log \left (\frac {4}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, {\left (d \tan \left (\frac {\pi \mathrm {sgn}\left (b\right )}{4 \, p} - \frac {\pi }{4 \, p}\right )^{2} + d\right )}} \]