\[ \int \frac {(e+f x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 \left (f x +e \right ) \arctanh \! \left ({\mathrm e}^{\mathrm {I} \left (d x +c \right )}\right )}{a d}-\frac {\left (f x +e \right ) \cot \! \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (f x +e \right ) \cot \! \left (d x +c \right )}{a d}+\frac {2 f \ln \! \left (\sin \! \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right )}{a \,d^{2}}+\frac {f \ln \! \left (\sin \! \left (d x +c \right )\right )}{a \,d^{2}}-\frac {\mathrm {I} f \polylog \! \left (2, -{\mathrm e}^{\mathrm {I} \left (d x +c \right )}\right )}{a \,d^{2}}+\frac {\mathrm {I} f \polylog \! \left (2, {\mathrm e}^{\mathrm {I} \left (d x +c \right )}\right )}{a \,d^{2}} \]
command
integrate((f*x+e)*csc(d*x+c)^2/(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
\[ \text {output too large to display} \]