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