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