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