18.7 Problem number 208

\[ \int \frac {\sin ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx \]

Optimal antiderivative \[ \frac {2}{5 b d \left (d \cos \! \left (b x +a \right )\right )^{\frac {5}{2}}}-\frac {2}{b \,d^{3} \sqrt {d \cos \! \left (b x +a \right )}} \]

command

integrate(sin(b*x+a)^3/(d*cos(b*x+a))^(7/2),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

\[ -\frac {2 \, {\left (5 \, b^{3} d^{3} \cos \left (b x + a\right )^{2} - b^{3} d^{3}\right )}}{5 \, \sqrt {d \cos \left (b x + a\right )} b^{4} d^{6} \cos \left (b x + a\right )^{2}} \]