18.6 Problem number 207

\[ \int \frac {\sin ^3(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx \]

Optimal antiderivative \[ \frac {2}{3 b d \left (d \cos \! \left (b x +a \right )\right )^{\frac {3}{2}}}+\frac {2 \sqrt {d \cos \! \left (b x +a \right )}}{b \,d^{3}} \]

command

integrate(sin(b*x+a)^3/(d*cos(b*x+a))^(5/2),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

\[ \frac {2 \, {\left (3 \, \sqrt {d \cos \left (b x + a\right )} b^{2} + \frac {b^{2} d}{\sqrt {d \cos \left (b x + a\right )} \cos \left (b x + a\right )}\right )}}{3 \, b^{3} d^{3}} \]