\[ \int \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (12 a^{2}-18 a b +5 b^{2}\right ) \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a -b}}\right )}{32 \left (a -b \right )^{\frac {3}{2}} d}+\frac {\left (12 a^{2}+18 a b +5 b^{2}\right ) \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right )}{32 \left (a +b \right )^{\frac {3}{2}} d}-\frac {\left (\sec ^{2}\left (d x +c \right )\right ) \left (a b -\left (6 a^{2}-5 b^{2}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{16 \left (a^{2}-b^{2}\right ) d}+\frac {\left (\sec ^{3}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}\, \tan \left (d x +c \right )}{4 d} \]
command
integrate(sec(d*x+c)^5*(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {b^{5} {\left (\frac {{\left (12 \, a^{2} - 18 \, a b + 5 \, b^{2}\right )} \arctan \left (\frac {\sqrt {b \sin \left (d x + c\right ) + a}}{\sqrt {-a + b}}\right )}{{\left (a b^{5} - b^{6}\right )} \sqrt {-a + b}} - \frac {{\left (12 \, a^{2} + 18 \, a b + 5 \, b^{2}\right )} \arctan \left (\frac {\sqrt {b \sin \left (d x + c\right ) + a}}{\sqrt {-a - b}}\right )}{{\left (a b^{5} + b^{6}\right )} \sqrt {-a - b}} - \frac {2 \, {\left (6 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2} - 18 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{3} + 18 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{4} - 6 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{5} - 5 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} b^{2} + 14 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a b^{2} - 23 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2} b^{2} + 14 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{3} b^{2} + 9 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} b^{4} - 8 \, \sqrt {b \sin \left (d x + c\right ) + a} a b^{4}\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} {\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{2} - 2 \, {\left (b \sin \left (d x + c\right ) + a\right )} a + a^{2} - b^{2}\right )}^{2}}\right )}}{32 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \sqrt {b \sin \left (d x + c\right ) + a} \sec \left (d x + c\right )^{5}\,{d x} \]________________________________________________________________________________________