\[ \int \frac {\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-1\right )^{\frac {2}{3}} b^{\frac {2}{3}} \arctan \left (\frac {\left (-1\right )^{\frac {1}{3}} b^{\frac {1}{3}}-a^{\frac {1}{3}} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {a^{\frac {2}{3}}-\left (-1\right )^{\frac {2}{3}} b^{\frac {2}{3}}}}\right )}{3 a^{\frac {2}{3}} \left (a^{\frac {2}{3}}-\left (-1\right )^{\frac {2}{3}} b^{\frac {2}{3}}\right )^{\frac {3}{2}} d}-\frac {2 b^{\frac {2}{3}} \arctan \left (\frac {b^{\frac {1}{3}}+a^{\frac {1}{3}} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {a^{\frac {2}{3}}-b^{\frac {2}{3}}}}\right )}{3 a^{\frac {2}{3}} \left (a^{\frac {2}{3}}-b^{\frac {2}{3}}\right )^{\frac {3}{2}} d}+\frac {2 \left (-1\right )^{\frac {1}{3}} b^{\frac {2}{3}} \arctan \left (\frac {\left (-1\right )^{\frac {2}{3}} b^{\frac {1}{3}}+a^{\frac {1}{3}} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {a^{\frac {2}{3}}+\left (-1\right )^{\frac {1}{3}} b^{\frac {2}{3}}}}\right )}{3 a^{\frac {2}{3}} \left (a^{\frac {2}{3}}+\left (-1\right )^{\frac {1}{3}} b^{\frac {2}{3}}\right )^{\frac {3}{2}} d}+\frac {\sec \left (d x +c \right ) \left (b -a \sin \left (d x +c \right )\right )}{\left (-a^{2}+b^{2}\right ) d} \]
command
integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]