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