\[ \int x \text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Optimal antiderivative \[ \left (\frac {1}{8}+\frac {i}{8}\right ) {\mathrm e}^{\frac {-2 a b \,d^{2} n \pi +2 i}{b^{2} d^{2} n^{2} \pi }} x^{2} \erf \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2}{n}+i a b \,d^{2} \pi +i b^{2} d^{2} \pi \ln \left (c \,x^{n}\right )\right )}{b d \sqrt {\pi }}\right ) \left (c \,x^{n}\right )^{-\frac {2}{n}}+\left (-\frac {1}{8}-\frac {i}{8}\right ) x^{2} \erfi \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2}{n}-i a b \,d^{2} \pi -i b^{2} d^{2} \pi \ln \left (c \,x^{n}\right )\right )}{b d \sqrt {\pi }}\right ) {\mathrm e}^{-\frac {2 \left (a b \,d^{2} n \pi +i\right )}{b^{2} d^{2} n^{2} \pi }} \left (c \,x^{n}\right )^{-\frac {2}{n}}+\frac {x^{2} \FresnelC \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2} \]
command
integrate(x*fresnel_cos(d*(a+b*log(c*x^n))),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{4} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {2 \, \log \left (c\right )}{n} - \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{4} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {2 \, \log \left (c\right )}{n} - \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \frac {1}{4} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {2 \, \log \left (c\right )}{n} - \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{4} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {2 \, \log \left (c\right )}{n} - \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \frac {1}{2} \, x^{2} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x {\rm fresnelc}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]