\[ \int x^2 \text {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Optimal antiderivative \[ \frac {x^{3} \cosineIntegral \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3}-\frac {x^{3} \expIntegral \left (\frac {\left (-i b d n +3\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\right ) {\mathrm e}^{-\frac {3 a}{b n}} \left (c \,x^{n}\right )^{-\frac {3}{n}}}{6}-\frac {x^{3} \expIntegral \left (\frac {\left (i b d n +3\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\right ) {\mathrm e}^{-\frac {3 a}{b n}} \left (c \,x^{n}\right )^{-\frac {3}{n}}}{6} \]
command
integrate(x^2*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
\[ \frac {1}{3} \, x^{3} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \frac {1}{6} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} - \frac {9 i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{6} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} + \frac {9 i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \frac {1}{6} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} - \frac {9 i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{6} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} + \frac {9 i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{2} \operatorname {Ci}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]