\[ \int \frac {(e+f x)^2 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {3 \,\mathrm {I} f \left (f x +e \right ) \polylog \! \left (2, \mathrm {I} \,{\mathrm e}^{\mathrm {I} \left (d x +c \right )}\right )}{4 a \,d^{2}}+\frac {5 f^{2} \arctanh \! \left (\sin \! \left (d x +c \right )\right )}{6 a \,d^{3}}+\frac {f^{2} \ln \! \left (\cos \! \left (d x +c \right )\right )}{3 a \,d^{3}}-\frac {3 \,\mathrm {I} \left (f x +e \right )^{2} \arctan \! \left ({\mathrm e}^{\mathrm {I} \left (d x +c \right )}\right )}{4 a d}+\frac {3 \,\mathrm {I} f \left (f x +e \right ) \polylog \! \left (2, \mathrm {-I} \,{\mathrm e}^{\mathrm {I} \left (d x +c \right )}\right )}{4 a \,d^{2}}-\frac {3 f^{2} \polylog \! \left (3, \mathrm {-I} \,{\mathrm e}^{\mathrm {I} \left (d x +c \right )}\right )}{4 a \,d^{3}}+\frac {3 f^{2} \polylog \! \left (3, \mathrm {I} \,{\mathrm e}^{\mathrm {I} \left (d x +c \right )}\right )}{4 a \,d^{3}}-\frac {3 f \left (f x +e \right ) \sec \! \left (d x +c \right )}{4 a \,d^{2}}-\frac {f^{2} \left (\sec ^{2}\left (d x +c \right )\right )}{12 a \,d^{3}}-\frac {f \left (f x +e \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{6 a \,d^{2}}-\frac {\left (f x +e \right )^{2} \left (\sec ^{4}\left (d x +c \right )\right )}{4 a d}+\frac {f \left (f x +e \right ) \tan \! \left (d x +c \right )}{3 a \,d^{2}}+\frac {f^{2} \sec \! \left (d x +c \right ) \tan \! \left (d x +c \right )}{12 a \,d^{3}}+\frac {3 \left (f x +e \right )^{2} \sec \! \left (d x +c \right ) \tan \! \left (d x +c \right )}{8 a d}+\frac {f \left (f x +e \right ) \left (\sec ^{2}\left (d x +c \right )\right ) \tan \! \left (d x +c \right )}{6 a \,d^{2}}+\frac {\left (f x +e \right )^{2} \left (\sec ^{3}\left (d x +c \right )\right ) \tan \! \left (d x +c \right )}{4 a d} \]
command
integrate((f*x+e)^2*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
\[ \text {output too large to display} \]