\[ \int \frac {(e+f x) \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {3 \,\mathrm {I} \left (f x +e \right ) \arctan \! \left ({\mathrm e}^{\mathrm {I} \left (d x +c \right )}\right )}{4 a d}+\frac {3 \,\mathrm {I} f \polylog \! \left (2, \mathrm {-I} \,{\mathrm e}^{\mathrm {I} \left (d x +c \right )}\right )}{8 a \,d^{2}}-\frac {3 \,\mathrm {I} f \polylog \! \left (2, \mathrm {I} \,{\mathrm e}^{\mathrm {I} \left (d x +c \right )}\right )}{8 a \,d^{2}}-\frac {3 f \sec \! \left (d x +c \right )}{8 a \,d^{2}}-\frac {f \left (\sec ^{3}\left (d x +c \right )\right )}{12 a \,d^{2}}-\frac {\left (f x +e \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{4 a d}+\frac {f \tan \! \left (d x +c \right )}{4 a \,d^{2}}+\frac {3 \left (f x +e \right ) \sec \! \left (d x +c \right ) \tan \! \left (d x +c \right )}{8 a d}+\frac {\left (f x +e \right ) \left (\sec ^{3}\left (d x +c \right )\right ) \tan \! \left (d x +c \right )}{4 a d}+\frac {f \left (\tan ^{3}\left (d x +c \right )\right )}{12 a \,d^{2}} \]
command
integrate((f*x+e)*sec(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \text {Exception raised: RuntimeError} \]
Maxima 5.44 via sagemath 9.3 output
\[ \text {output too large to display} \]