\[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {\left (2 a -3 b \right ) \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a -b}}\right )}{4 \left (a -b \right )^{\frac {3}{2}} d}+\frac {\left (2 a +3 b \right ) \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right )}{4 \left (a +b \right )^{\frac {3}{2}} d}-\frac {\left (\sec ^{2}\left (d x +c \right )\right ) \left (b -a \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{2 \left (a^{2}-b^{2}\right ) d} \]
command
integrate(sec(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {b^{3} {\left (\frac {{\left (2 \, a - 3 \, b\right )} \arctan \left (\frac {\sqrt {b \sin \left (d x + c\right ) + a}}{\sqrt {-a + b}}\right )}{{\left (a b^{3} - b^{4}\right )} \sqrt {-a + b}} - \frac {{\left (2 \, a + 3 \, b\right )} \arctan \left (\frac {\sqrt {b \sin \left (d x + c\right ) + a}}{\sqrt {-a - b}}\right )}{{\left (a b^{3} + b^{4}\right )} \sqrt {-a - b}} - \frac {2 \, {\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a - \sqrt {b \sin \left (d x + c\right ) + a} a^{2} - \sqrt {b \sin \left (d x + c\right ) + a} b^{2}\right )}}{{\left (a^{2} b^{2} - b^{4}\right )} {\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{2} - 2 \, {\left (b \sin \left (d x + c\right ) + a\right )} a + a^{2} - b^{2}\right )}}\right )}}{4 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sec \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right ) + a}}\,{d x} \]________________________________________________________________________________________