\[ \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \sin \left (b x +a \right )}{5 b d \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}}}-\frac {4 \sin \left (b x +a \right )}{5 b \,d^{3} \sqrt {d \cos \left (b x +a \right )}}+\frac {4 \sqrt {\frac {\cos \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right ) \sqrt {d \cos \left (b x +a \right )}}{5 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) b \,d^{4} \sqrt {\cos \left (b x +a \right )}} \]
command
integrate(sin(b*x+a)^2/(d*cos(b*x+a))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (-i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {d \cos \left (b x + a\right )} {\left (2 \, \cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )\right )}}{5 \, b d^{4} \cos \left (b x + a\right )^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {d \cos \left (b x + a\right )} {\left (\cos \left (b x + a\right )^{2} - 1\right )}}{d^{4} \cos \left (b x + a\right )^{4}}, x\right ) \]