\[ \int (e x)^m \text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Optimal antiderivative \[ \frac {\left (\frac {1}{4}+\frac {i}{4}\right ) {\mathrm e}^{\frac {i \left (1+m \right ) \left (2 i a b \,d^{2} n \pi +m +1\right )}{2 b^{2} d^{2} n^{2} \pi }} x \left (e x \right )^{m} \erf \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+m +i a b \,d^{2} n \pi +i b^{2} d^{2} n \pi \ln \left (c \,x^{n}\right )\right )}{b d n \sqrt {\pi }}\right ) \left (c \,x^{n}\right )^{-\frac {1+m}{n}}}{1+m}+\frac {\left (-\frac {1}{4}-\frac {i}{4}\right ) x \left (e x \right )^{m} \erfi \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+m -i a b \,d^{2} n \pi -i b^{2} d^{2} n \pi \ln \left (c \,x^{n}\right )\right )}{b d n \sqrt {\pi }}\right ) {\mathrm e}^{-\frac {i \left (1+m \right ) \left (-2 i a b \,d^{2} n \pi +m +1\right )}{2 b^{2} d^{2} n^{2} \pi }} \left (c \,x^{n}\right )^{-\frac {1+m}{n}}}{1+m}+\frac {\left (e x \right )^{1+m} \FresnelC \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{e \left (1+m \right )} \]
command
integrate((e*x)^m*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 {\pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} - \frac {i \, m}{\pi b^{2} d^{2} n^{2}} - \frac {i}{2 \, \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 + i \, m + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} + \frac {i \, m}{\pi b^{2} d^{2} n^{2}} + \frac {i}{2 \, \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 - i \, m - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} - \frac {i \, m}{\pi b^{2} d^{2} n^{2}} - \frac {i}{2 \, \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 + i \, m + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} + \frac {i \, m}{\pi b^{2} d^{2} n^{2}} + \frac {i}{2 \, \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 - i \, m - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - 2 \, x e^{\left (m \log \left (x\right ) + m\right )} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right )}{2 \, {\left (m + 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\left (e x\right )^{m} {\rm fresnelc}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]