\[ \int \frac {\sec ^5(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {3 \left (4 a^{2}-10 a b +7 b^{2}\right ) \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a -b}}\right )}{32 \left (a -b \right )^{\frac {5}{2}} d}+\frac {3 \left (4 a^{2}+10 a b +7 b^{2}\right ) \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right )}{32 \left (a +b \right )^{\frac {5}{2}} d}-\frac {\left (\sec ^{4}\left (d x +c \right )\right ) \left (b -a \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{4 \left (a^{2}-b^{2}\right ) d}-\frac {\left (\sec ^{2}\left (d x +c \right )\right ) \left (b \left (a^{2}-7 b^{2}\right )-6 a \left (a^{2}-2 b^{2}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{16 \left (a^{2}-b^{2}\right )^{2} d} \]
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
\[ \frac {b^{5} {\left (\frac {3 \, {\left (4 \, a^{2} - 10 \, a b + 7 \, b^{2}\right )} \arctan \left (\frac {\sqrt {b \sin \left (d x + c\right ) + a}}{\sqrt {-a + b}}\right )}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} \sqrt {-a + b}} - \frac {3 \, {\left (4 \, a^{2} + 10 \, a b + 7 \, b^{2}\right )} \arctan \left (\frac {\sqrt {b \sin \left (d x + c\right ) + a}}{\sqrt {-a - b}}\right )}{{\left (a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right )} \sqrt {-a - b}} - \frac {2 \, {\left (6 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{3} - 18 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{4} + 18 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{5} - 6 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{6} - 12 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a b^{2} + 35 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} b^{2} - 44 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} b^{2} + 21 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{4} b^{2} + 7 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} b^{4} + 2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a b^{4} - 4 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{2} b^{4} - 11 \, \sqrt {b \sin \left (d x + c\right ) + a} b^{6}\right )}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} {\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{2} - 2 \, {\left (b \sin \left (d x + c\right ) + a\right )} a + a^{2} - b^{2}\right )}^{2}}\right )}}{32 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sec \left (d x + c\right )^{5}}{\sqrt {b \sin \left (d x + c\right ) + a}}\,{d x} \]________________________________________________________________________________________