\[ \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {2 \arcsinh \left (\frac {\sqrt {e \cos \left (d x +c \right )}}{\sqrt {e}}\right ) \sqrt {1+\cos \left (d x +c \right )}\, \sqrt {a +a \sin \left (d x +c \right )}}{d \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {e}}+\frac {2 \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {e}}{\sqrt {e \cos \left (d x +c \right )}\, \sqrt {1+\cos \left (d x +c \right )}}\right ) \sqrt {1+\cos \left (d x +c \right )}\, \sqrt {a +a \sin \left (d x +c \right )}}{d \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {e}} \]
command
integrate((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]