\[ \int x^m \text {CosIntegral}(b x) \, dx \]
Optimal antiderivative \[ \frac {x^{1+m} \cosineIntegral \left (b x \right )}{1+m}+\frac {i x^{m} \Gamma \left (1+m , -i b x \right ) \left (-i b x \right )^{-m}}{2 b \left (1+m \right )}-\frac {i x^{m} \Gamma \left (1+m , i b x \right ) \left (i b x \right )^{-m}}{2 b \left (1+m \right )} \]
command
integrate(x^m*fresnel_cos(b*x),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, \pi b x x^{m} \operatorname {C}\left (b x\right ) - i \, {\left (\cosh \left (\frac {1}{2} \, m \log \left (\frac {1}{2} i \, \pi b^{2}\right )\right ) - \sinh \left (\frac {1}{2} \, m \log \left (\frac {1}{2} i \, \pi b^{2}\right )\right )\right )} \Gamma \left (\frac {1}{2} \, m + 1, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + i \, {\left (\cosh \left (\frac {1}{2} \, m \log \left (-\frac {1}{2} i \, \pi b^{2}\right )\right ) - \sinh \left (\frac {1}{2} \, m \log \left (-\frac {1}{2} i \, \pi b^{2}\right )\right )\right )} \Gamma \left (\frac {1}{2} \, m + 1, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )}{2 \, \pi {\left (b m + b\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{m} \operatorname {Ci}\left (b x\right ), x\right ) \]