\[ \int \frac {\cos ^4(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \cos \left (d x +c \right )}}\right ) \sqrt {2}}{d \sqrt {a}}-\frac {148 \sin \left (d x +c \right )}{105 d \sqrt {a +a \cos \left (d x +c \right )}}-\frac {2 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{7 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {62 \sin \left (d x +c \right ) \sqrt {a +a \cos \left (d x +c \right )}}{105 a d} \]
command
integrate(cos(d*x+c)^4/(a+a*cos(d*x+c))^(1/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \text {output too large to display} \]
Maxima 5.44 via sagemath 9.3 output \[ \text {Timed out} \]_____________________________________________________