\[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (A -B \right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sin \! \left (d x +c \right )}{2 d \left (a +a \cos \! \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\left (15 A -19 B \right ) \arctanh \! \left (\frac {\sin \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \cos \left (d x +c \right )}}\right ) \sqrt {2}}{4 a^{\frac {3}{2}} d}+\frac {\left (651 A -799 B \right ) \sin \! \left (d x +c \right )}{105 a d \sqrt {a +a \cos \! \left (d x +c \right )}}+\frac {\left (63 A -67 B \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \! \left (d x +c \right )}{70 a d \sqrt {a +a \cos \! \left (d x +c \right )}}-\frac {\left (7 A -11 B \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sin \! \left (d x +c \right )}{14 a d \sqrt {a +a \cos \! \left (d x +c \right )}}-\frac {\left (273 A -397 B \right ) \sin \! \left (d x +c \right ) \sqrt {a +a \cos \! \left (d x +c \right )}}{210 a^{2} d} \]
command
integrate(cos(d*x+c)^4*(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
\[ \frac {\frac {105 \, {\left (15 \, \sqrt {2} A - 19 \, \sqrt {2} B\right )} \log \left ({\left | -\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \right |}\right )}{a^{\frac {3}{2}}} + \frac {{\left ({\left ({\left ({\left (\frac {105 \, {\left (\sqrt {2} A a^{5} - \sqrt {2} B a^{5}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{3}} + \frac {4 \, {\left (693 \, \sqrt {2} A a^{5} - 877 \, \sqrt {2} B a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {14 \, {\left (453 \, \sqrt {2} A a^{5} - 517 \, \sqrt {2} B a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {140 \, {\left (39 \, \sqrt {2} A a^{5} - 47 \, \sqrt {2} B a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1785 \, {\left (\sqrt {2} A a^{5} - \sqrt {2} B a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {7}{2}}}}{420 \, d} \]