\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x) \, dx \]
Optimal antiderivative \[ \frac {105 x^{2}}{4 b^{7} \pi ^{4}}-\frac {7 x^{6}}{12 b^{3} \pi ^{2}}-\frac {55 x^{2} \cos \left (b^{2} \pi \,x^{2}\right )}{4 b^{7} \pi ^{4}}+\frac {x^{6} \cos \left (b^{2} \pi \,x^{2}\right )}{4 b^{3} \pi ^{2}}-\frac {105 x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{b^{8} \pi ^{4}}+\frac {7 x^{5} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{b^{4} \pi ^{2}}+\frac {105 \FresnelC \left (b x \right )^{2}}{2 b^{9} \pi ^{4}}-\frac {35 x^{3} \FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{6} \pi ^{3}}+\frac {x^{7} \FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {40 \sin \left (b^{2} \pi \,x^{2}\right )}{b^{9} \pi ^{5}}-\frac {5 x^{4} \sin \left (b^{2} \pi \,x^{2}\right )}{2 b^{5} \pi ^{3}} \]
command
integrate(x^8*cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, \pi ^{3} b^{6} x^{6} - 240 \, \pi b^{2} x^{2} - 3 \, {\left (\pi ^{3} b^{6} x^{6} - 55 \, \pi b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 42 \, {\left (\pi ^{3} b^{5} x^{5} - 15 \, \pi b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 315 \, \pi \operatorname {C}\left (b x\right )^{2} + 6 \, {\left (5 \, {\left (\pi ^{2} b^{4} x^{4} - 16\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{4} b^{7} x^{7} - 35 \, \pi ^{2} b^{3} x^{3}\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{6 \, \pi ^{5} b^{9}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right ), x\right ) \]