\[ \int x^6 \text {FresnelC}(b x) \, dx \]
Optimal antiderivative \[ \frac {48 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 b^{7} \pi ^{4}}-\frac {6 x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 b^{3} \pi ^{2}}+\frac {x^{7} \FresnelC \left (b x \right )}{7}+\frac {24 x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 b^{5} \pi ^{3}}-\frac {x^{6} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 b \pi } \]
command
integrate(x^6*fresnel_cos(b*x),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{7} \, x^{7} \operatorname {C}\left (b x\right ) - \frac {6 \, {\left (\pi ^{2} b^{4} x^{4} - 8\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{3} b^{6} x^{6} - 24 \, \pi b^{2} x^{2}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{7 \, \pi ^{4} b^{7}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int x^{6} {\rm fresnelc}\left (b x\right )\,{d x} \]________________________________________________________________________________________