\[ \int \frac {\text {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx \]
Optimal antiderivative \[ \frac {\cosineIntegral \! \left (d \left (a +b \ln \! \left (c \,x^{n}\right )\right )\right ) \left (a +b \ln \! \left (c \,x^{n}\right )\right )}{b n}-\frac {\sin \! \left (d \left (a +b \ln \! \left (c \,x^{n}\right )\right )\right )}{b d n} \]
command
integrate(fresnel_cos(d*(a+b*log(c*x^n)))/x,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \operatorname {Ci}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) - \sin \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}{b d n} \]