\[ \int x \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Optimal antiderivative \[ -\frac {i x^{2} \expIntegral \left (\frac {\left (-i b d n +2\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\right ) {\mathrm e}^{-\frac {2 a}{b n}} \left (c \,x^{n}\right )^{-\frac {2}{n}}}{4}+\frac {i x^{2} \expIntegral \left (\frac {\left (i b d n +2\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\right ) {\mathrm e}^{-\frac {2 a}{b n}} \left (c \,x^{n}\right )^{-\frac {2}{n}}}{4}+\frac {x^{2} \sinIntegral \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2} \]
command
integrate(x*sin_integral(d*(a+b*log(c*x^n))),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{2} \, x^{2} \operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \frac {1}{4} \, {\left (i \, {\rm Ei}\left (\frac {i \, a b d n + {\left (i \, b^{2} d n + 2 \, b\right )} \log \left (c\right ) + {\left (i \, b^{2} d n^{2} + 2 \, b n\right )} \log \left (x\right ) + 2 \, a}{b n}\right ) - i \, {\rm Ei}\left (\frac {-i \, a b d n + {\left (-i \, b^{2} d n + 2 \, b\right )} \log \left (c\right ) + {\left (-i \, b^{2} d n^{2} + 2 \, b n\right )} \log \left (x\right ) + 2 \, a}{b n}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x \operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]