\[ \int \frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx \]
Optimal antiderivative \[ \frac {\cos \left (\frac {d^{2} \pi \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{2}\right )}{b d n \pi }+\frac {\mathrm {S}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n} \]
command
integrate(fresnel_sin(d*(a+b*log(c*x^n)))/x,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (\pi b d n \log \left (x\right ) + \pi b d \log \left (c\right ) + \pi a d\right )} \operatorname {S}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \cos \left (\frac {1}{2} \, \pi b^{2} d^{2} n^{2} \log \left (x\right )^{2} + \pi b^{2} d^{2} n \log \left (c\right ) \log \left (x\right ) + \frac {1}{2} \, \pi b^{2} d^{2} \log \left (c\right )^{2} + \pi a b d^{2} n \log \left (x\right ) + \pi a b d^{2} \log \left (c\right ) + \frac {1}{2} \, \pi a^{2} d^{2}\right )}{\pi b d n} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\rm fresnels}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x}, x\right ) \]