\[ \int \sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (2 a -b \right ) \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a -b}}\right )}{4 d \sqrt {a -b}}+\frac {\left (2 a +b \right ) \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right )}{4 d \sqrt {a +b}}+\frac {\sec \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, \tan \left (d x +c \right )}{2 d} \]
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
\[ \frac {b^{3} {\left (\frac {{\left (2 \, a - b\right )} \arctan \left (\frac {\sqrt {b \sin \left (d x + c\right ) + a}}{\sqrt {-a + b}}\right )}{\sqrt {-a + b} b^{3}} - \frac {{\left (2 \, a + b\right )} \arctan \left (\frac {\sqrt {b \sin \left (d x + c\right ) + a}}{\sqrt {-a - b}}\right )}{\sqrt {-a - b} b^{3}} - \frac {2 \, {\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} - \sqrt {b \sin \left (d x + c\right ) + a} a\right )}}{{\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{2} - 2 \, {\left (b \sin \left (d x + c\right ) + a\right )} a + a^{2} - b^{2}\right )} b^{2}}\right )}}{4 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \sqrt {b \sin \left (d x + c\right ) + a} \sec \left (d x + c\right )^{3}\,{d x} \]________________________________________________________________________________________