\[ \int \frac {\text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx \]
Optimal antiderivative \[ \frac {\FresnelC \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}-\frac {\sin \left (\frac {d^{2} \pi \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{2}\right )}{b d n \pi } \]
command
integrate(fresnel_cos(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) - \frac {\sin \left (\frac {1}{2} \, \pi b^{2} d^{2} \log \left (c x^{n}\right )^{2} + \pi a b d^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, \pi a^{2} d^{2}\right )}{\pi }}{b d n} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {{\rm fresnelc}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x}\,{d x} \]________________________________________________________________________________________