\[ \int \sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {\frac {\cos \left (d +e x -\arctan \left (c , a\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (c , a\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {a^{2}+c^{2}}}{b +\sqrt {a^{2}+c^{2}}}}\right ) \sqrt {a +c \cot \left (e x +d \right )+b \csc \left (e x +d \right )}\, \left (\sqrt {\sin }\left (e x +d \right )\right )}{\cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (c , a\right )}{2}\right ) e \sqrt {\frac {b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )}{b +\sqrt {a^{2}+c^{2}}}}} \]
command
integrate((a+c*cot(e*x+d)+b*csc(e*x+d))^(1/2)*sin(e*x+d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (\sqrt {2} {\left (a b + i \, b c\right )} \sqrt {i \, a + c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{2} c^{2} + 6 i \, a c^{3} - 3 \, c^{4} + 2 i \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} c\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (-9 i \, a^{5} b + 8 i \, a^{3} b^{3} + 27 i \, a b c^{4} - 9 \, b c^{5} + 2 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{3} + 6 i \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} c^{2} + 3 \, {\left (9 \, a^{4} b - 8 \, a^{2} b^{3}\right )} c\right )}}{27 \, {\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )}}, \frac {-2 i \, a b + 2 \, b c + 3 \, {\left (a^{2} + c^{2}\right )} \cos \left (x e + d\right ) - 3 \, {\left (i \, a^{2} + i \, c^{2}\right )} \sin \left (x e + d\right )}{3 \, {\left (a^{2} + c^{2}\right )}}\right ) + \sqrt {2} {\left (a b - i \, b c\right )} \sqrt {-i \, a + c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{2} c^{2} - 6 i \, a c^{3} - 3 \, c^{4} - 2 i \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} c\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (9 i \, a^{5} b - 8 i \, a^{3} b^{3} - 27 i \, a b c^{4} - 9 \, b c^{5} + 2 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{3} - 6 i \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} c^{2} + 3 \, {\left (9 \, a^{4} b - 8 \, a^{2} b^{3}\right )} c\right )}}{27 \, {\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )}}, \frac {2 i \, a b + 2 \, b c + 3 \, {\left (a^{2} + c^{2}\right )} \cos \left (x e + d\right ) - 3 \, {\left (-i \, a^{2} - i \, c^{2}\right )} \sin \left (x e + d\right )}{3 \, {\left (a^{2} + c^{2}\right )}}\right ) - 3 \, \sqrt {2} {\left (i \, a^{2} + i \, c^{2}\right )} \sqrt {i \, a + c} {\rm weierstrassZeta}\left (\frac {4 \, {\left (3 \, a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{2} c^{2} + 6 i \, a c^{3} - 3 \, c^{4} + 2 i \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} c\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (-9 i \, a^{5} b + 8 i \, a^{3} b^{3} + 27 i \, a b c^{4} - 9 \, b c^{5} + 2 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{3} + 6 i \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} c^{2} + 3 \, {\left (9 \, a^{4} b - 8 \, a^{2} b^{3}\right )} c\right )}}{27 \, {\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{2} c^{2} + 6 i \, a c^{3} - 3 \, c^{4} + 2 i \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} c\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (-9 i \, a^{5} b + 8 i \, a^{3} b^{3} + 27 i \, a b c^{4} - 9 \, b c^{5} + 2 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{3} + 6 i \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} c^{2} + 3 \, {\left (9 \, a^{4} b - 8 \, a^{2} b^{3}\right )} c\right )}}{27 \, {\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )}}, \frac {-2 i \, a b + 2 \, b c + 3 \, {\left (a^{2} + c^{2}\right )} \cos \left (x e + d\right ) - 3 \, {\left (i \, a^{2} + i \, c^{2}\right )} \sin \left (x e + d\right )}{3 \, {\left (a^{2} + c^{2}\right )}}\right )\right ) - 3 \, \sqrt {2} {\left (-i \, a^{2} - i \, c^{2}\right )} \sqrt {-i \, a + c} {\rm weierstrassZeta}\left (\frac {4 \, {\left (3 \, a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{2} c^{2} - 6 i \, a c^{3} - 3 \, c^{4} - 2 i \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} c\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (9 i \, a^{5} b - 8 i \, a^{3} b^{3} - 27 i \, a b c^{4} - 9 \, b c^{5} + 2 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{3} - 6 i \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} c^{2} + 3 \, {\left (9 \, a^{4} b - 8 \, a^{2} b^{3}\right )} c\right )}}{27 \, {\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{2} c^{2} - 6 i \, a c^{3} - 3 \, c^{4} - 2 i \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} c\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (9 i \, a^{5} b - 8 i \, a^{3} b^{3} - 27 i \, a b c^{4} - 9 \, b c^{5} + 2 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{3} - 6 i \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} c^{2} + 3 \, {\left (9 \, a^{4} b - 8 \, a^{2} b^{3}\right )} c\right )}}{27 \, {\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )}}, \frac {2 i \, a b + 2 \, b c + 3 \, {\left (a^{2} + c^{2}\right )} \cos \left (x e + d\right ) - 3 \, {\left (-i \, a^{2} - i \, c^{2}\right )} \sin \left (x e + d\right )}{3 \, {\left (a^{2} + c^{2}\right )}}\right )\right )\right )} e^{\left (-1\right )}}{3 \, {\left (a^{2} + c^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a} \sqrt {\sin \left (e x + d\right )}, x\right ) \]