\[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx \]
Optimal antiderivative \[ -\frac {i {\mathrm e}^{\frac {2 a}{b n}} \left (c \,x^{n}\right )^{\frac {2}{n}} \expIntegral \left (-\frac {\left (-i b d n +2\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\right )}{4 x^{2}}+\frac {i {\mathrm e}^{\frac {2 a}{b n}} \left (c \,x^{n}\right )^{\frac {2}{n}} \expIntegral \left (-\frac {\left (i b d n +2\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\right )}{4 x^{2}}-\frac {\sinIntegral \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2 x^{2}} \]
command
integrate(sin_integral(d*(a+b*log(c*x^n)))/x^3,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (-i \, x^{2} {\rm Ei}\left (\frac {i \, a b d n + {\left (i \, b^{2} d n - 2 \, b\right )} \log \left (c\right ) + {\left (i \, b^{2} d n^{2} - 2 \, b n\right )} \log \left (x\right ) - 2 \, a}{b n}\right ) + i \, x^{2} {\rm Ei}\left (\frac {-i \, a b d n + {\left (-i \, b^{2} d n - 2 \, b\right )} \log \left (c\right ) + {\left (-i \, b^{2} d n^{2} - 2 \, b n\right )} \log \left (x\right ) - 2 \, a}{b n}\right )\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 2 \, \operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right )}{4 \, x^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{3}}, x\right ) \]