\[ \int \frac {\cot (c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \! \left (\frac {\sqrt {a +b \left (\sin ^{4}\left (d x +c \right )\right )}}{\sqrt {a}}\right )}{2 d \sqrt {a}} \]
command
integrate(cot(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Timed out} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {\arctan \left (\frac {\sqrt {b \sin \left (d x + c\right )^{4} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} d} \]