\[ \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {12 \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}} \sin \left (b x +a \right )}{5 b \,d^{3}}+\frac {2 \left (\sin ^{3}\left (b x +a \right )\right )}{b d \sqrt {d \cos \left (b x +a \right )}}-\frac {24 \sqrt {\frac {\cos \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right ) \sqrt {d \cos \left (b x +a \right )}}{5 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) b \,d^{2} \sqrt {\cos \left (b x +a \right )}} \]
command
integrate(sin(b*x+a)^4/(d*cos(b*x+a))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (6 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - 6 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - \sqrt {d \cos \left (b x + a\right )} {\left (\cos \left (b x + a\right )^{2} + 5\right )} \sin \left (b x + a\right )\right )}}{5 \, b d^{2} \cos \left (b x + a\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt {d \cos \left (b x + a\right )}}{d^{2} \cos \left (b x + a\right )^{2}}, x\right ) \]