\[ \int \frac {\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (A -B \right ) \sin \! \left (d x +c \right )}{2 d \left (a +a \cos \! \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {\left (3 A -7 B \right ) \arctanh \! \left (\frac {\sin \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \cos \left (d x +c \right )}}\right ) \sqrt {2}}{4 a^{\frac {3}{2}} d}+\frac {2 B \sin \! \left (d x +c \right )}{a d \sqrt {a +a \cos \! \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {\frac {{\left (\frac {\sqrt {2} {\left (A a^{2} - B a^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{3}} + \frac {\sqrt {2} {\left (A a^{2} - 9 \, B a^{2}\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}} + \frac {\sqrt {2} {\left (3 \, A - 7 \, B\right )} \log \left ({\left | -\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \right |}\right )}{a^{\frac {3}{2}}}}{4 \, d} \]