\[ \int \frac {\sin (c+d x)}{x (a+b x)^3} \, dx \]
Optimal antiderivative \[ -\frac {d \cosineIntegral \left (\frac {a d}{b}+d x \right ) \cos \left (-c +\frac {a d}{b}\right )}{a^{2} b}+\frac {d \cos \left (d x +c \right )}{2 a b \left (b x +a \right )}+\frac {\cos \left (c \right ) \sinIntegral \left (d x \right )}{a^{3}}-\frac {\cos \left (-c +\frac {a d}{b}\right ) \sinIntegral \left (\frac {a d}{b}+d x \right )}{a^{3}}+\frac {d^{2} \cos \left (-c +\frac {a d}{b}\right ) \sinIntegral \left (\frac {a d}{b}+d x \right )}{2 a \,b^{2}}+\frac {\cosineIntegral \left (d x \right ) \sin \left (c \right )}{a^{3}}+\frac {\cosineIntegral \left (\frac {a d}{b}+d x \right ) \sin \left (-c +\frac {a d}{b}\right )}{a^{3}}-\frac {d^{2} \cosineIntegral \left (\frac {a d}{b}+d x \right ) \sin \left (-c +\frac {a d}{b}\right )}{2 a \,b^{2}}-\frac {d \sinIntegral \left (\frac {a d}{b}+d x \right ) \sin \left (-c +\frac {a d}{b}\right )}{a^{2} b}+\frac {\sin \left (d x +c \right )}{2 a \left (b x +a \right )^{2}}+\frac {\sin \left (d x +c \right )}{a^{2} \left (b x +a \right )} \]
command
integrate(sin(d*x+c)/x/(b*x+a)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________