\[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (b \cos \left (d x +c \right )-a \sin \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) d \sqrt {a \cos \left (d x +c \right )+b \sin \left (d x +c \right )}}-\frac {2 \sqrt {\frac {\cos \left (c +d x -\arctan \left (a , b\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (a , b\right )}{2}\right ), \sqrt {2}\right ) \sqrt {a \cos \left (d x +c \right )+b \sin \left (d x +c \right )}}{\cos \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (a , b\right )}{2}\right ) \left (a^{2}+b^{2}\right ) d \sqrt {\frac {a \cos \left (d x +c \right )+b \sin \left (d x +c \right )}{\sqrt {a^{2}+b^{2}}}}} \]
command
integrate(1/(a*cos(d*x+c)+b*sin(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (-i \, \sqrt {2} a \cos \left (d x + c\right ) - i \, \sqrt {2} b \sin \left (d x + c\right )\right )} \sqrt {a - i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a^{2} + 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a^{2} + 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + {\left (i \, \sqrt {2} a \cos \left (d x + c\right ) + i \, \sqrt {2} b \sin \left (d x + c\right )\right )} \sqrt {a + i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a^{2} - 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a^{2} - 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, \sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{{\left (a^{3} + a b^{2}\right )} d \cos \left (d x + c\right ) + {\left (a^{2} b + b^{3}\right )} d \sin \left (d x + c\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}, x\right ) \]