\[ \int \frac {\sec ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (a +7 b \right ) \ln \left (1-\sin \left (d x +c \right )\right )}{4 \left (a +b \right )^{3} d}+\frac {\left (a -7 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{3} d}+\frac {b^{\frac {5}{3}} \left (4 a^{2}+3 a^{\frac {4}{3}} b^{\frac {2}{3}}+2 b^{2}\right ) \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} \sin \left (d x +c \right )\right )}{9 a^{\frac {5}{3}} \left (a^{2}-b^{2}\right )^{2} d}+\frac {b^{\frac {5}{3}} \left (3 b^{\frac {2}{3}} \left (3 a^{2}+b^{2}\right )+4 a^{\frac {2}{3}} \left (a^{2}+2 b^{2}\right )\right ) \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} \sin \left (d x +c \right )\right )}{3 a^{\frac {1}{3}} \left (a^{2}-b^{2}\right )^{3} d}-\frac {b^{\frac {5}{3}} \left (4 a^{2}+3 a^{\frac {4}{3}} b^{\frac {2}{3}}+2 b^{2}\right ) \ln \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} \sin \left (d x +c \right )+b^{\frac {2}{3}} \left (\sin ^{2}\left (d x +c \right )\right )\right )}{18 a^{\frac {5}{3}} \left (a^{2}-b^{2}\right )^{2} d}-\frac {b^{\frac {5}{3}} \left (3 b^{\frac {2}{3}} \left (3 a^{2}+b^{2}\right )+4 a^{\frac {2}{3}} \left (a^{2}+2 b^{2}\right )\right ) \ln \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} \sin \left (d x +c \right )+b^{\frac {2}{3}} \left (\sin ^{2}\left (d x +c \right )\right )\right )}{6 a^{\frac {1}{3}} \left (a^{2}-b^{2}\right )^{3} d}+\frac {2 a b \left (a^{2}+5 b^{2}\right ) \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3 \left (a^{2}-b^{2}\right )^{3} d}+\frac {1}{4 \left (a +b \right )^{2} d \left (1-\sin \left (d x +c \right )\right )}-\frac {1}{4 \left (a -b \right )^{2} d \left (1+\sin \left (d x +c \right )\right )}-\frac {b \left (a \left (a^{2}+2 b^{2}\right )-b \sin \left (d x +c \right ) \left (2 a^{2}+b^{2}-3 a b \sin \left (d x +c \right )\right )\right )}{3 a \left (a^{2}-b^{2}\right )^{2} d \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}-\frac {b^{\frac {5}{3}} \left (4 a^{2}-3 a^{\frac {4}{3}} b^{\frac {2}{3}}+2 b^{2}\right ) \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} \sin \left (d x +c \right )\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{9 a^{\frac {5}{3}} \left (a^{2}-b^{2}\right )^{2} d}-\frac {b^{\frac {5}{3}} \left (4 a^{\frac {8}{3}}-9 a^{2} b^{\frac {2}{3}}+8 a^{\frac {2}{3}} b^{2}-3 b^{\frac {8}{3}}\right ) \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} \sin \left (d x +c \right )\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}} \left (a^{2}-b^{2}\right )^{3} d} \]
command
integrate(sec(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,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 \[ \text {Timed out} \]