\[ \int \frac {\sin (c+d x)}{x^2 (a+b x)^3} \, dx \]
Optimal antiderivative \[ \frac {d \cosineIntegral \left (d x \right ) \cos \left (c \right )}{a^{3}}+\frac {2 d \cosineIntegral \left (\frac {a d}{b}+d x \right ) \cos \left (-c +\frac {a d}{b}\right )}{a^{3}}-\frac {d \cos \left (d x +c \right )}{2 a^{2} \left (b x +a \right )}-\frac {3 b \cos \left (c \right ) \sinIntegral \left (d x \right )}{a^{4}}+\frac {3 b \cos \left (-c +\frac {a d}{b}\right ) \sinIntegral \left (\frac {a d}{b}+d x \right )}{a^{4}}-\frac {d^{2} \cos \left (-c +\frac {a d}{b}\right ) \sinIntegral \left (\frac {a d}{b}+d x \right )}{2 a^{2} b}-\frac {3 b \cosineIntegral \left (d x \right ) \sin \left (c \right )}{a^{4}}-\frac {d \sinIntegral \left (d x \right ) \sin \left (c \right )}{a^{3}}-\frac {3 b \cosineIntegral \left (\frac {a d}{b}+d x \right ) \sin \left (-c +\frac {a d}{b}\right )}{a^{4}}+\frac {d^{2} \cosineIntegral \left (\frac {a d}{b}+d x \right ) \sin \left (-c +\frac {a d}{b}\right )}{2 a^{2} b}+\frac {2 d \sinIntegral \left (\frac {a d}{b}+d x \right ) \sin \left (-c +\frac {a d}{b}\right )}{a^{3}}-\frac {\sin \left (d x +c \right )}{a^{3} x}-\frac {b \sin \left (d x +c \right )}{2 a^{2} \left (b x +a \right )^{2}}-\frac {2 b \sin \left (d x +c \right )}{a^{3} \left (b x +a \right )} \]
command
integrate(sin(d*x+c)/x^2/(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 \[ \text {Timed out} \]_______________________________________________________