47.100 Problem number 269

\[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx \]

Optimal antiderivative \[ \frac {4 \cos \left (b x +a \right )}{15 b \csc \left (b x +a \right )^{\frac {3}{2}}}+\frac {2 \left (\cos ^{3}\left (b x +a \right )\right )}{9 b \csc \left (b x +a \right )^{\frac {3}{2}}}-\frac {8 \sqrt {\frac {1}{2}+\frac {\sin \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\csc }\left (b x +a \right )\right ) \left (\sqrt {\sin }\left (b x +a \right )\right )}{15 \sin \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b x}{2}\right ) b} \]

command

integrate(cos(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

\[ \frac {2 \, {\left (6 \, \sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 6 \, \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - \frac {5 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3} - 6 \, \cos \left (b x + a\right )}{\sqrt {\sin \left (b x + a\right )}}\right )}}{45 \, b} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (\frac {\cos \left (b x + a\right )^{4}}{\sqrt {\csc \left (b x + a\right )}}, x\right ) \]