\[ \int \frac {\sin ^3(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (d \cos \! \left (b x +a \right )\right )^{\frac {3}{2}}}{3 b \,d^{3}}+\frac {2}{b d \sqrt {d \cos \! \left (b x +a \right )}} \]
command
integrate(sin(b*x+a)^3/(d*cos(b*x+a))^(3/2),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
\[ \frac {2 \, {\left (\frac {\sqrt {d \cos \left (b x + a\right )} \cos \left (b x + a\right )}{d} + \frac {3}{\sqrt {d \cos \left (b x + a\right )}}\right )}}{3 \, b d} \]