\[ \int x^7 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \]
Optimal antiderivative \[ -\frac {4 x^{3}}{b^{5} \pi ^{3}}+\frac {x^{7}}{14 b \pi }+\frac {17 x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{8 b^{5} \pi ^{3}}+\frac {24 x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{b^{6} \pi ^{3}}-\frac {x^{6} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{b^{2} \pi }-\frac {48 \FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{8} \pi ^{4}}+\frac {6 x^{4} \FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{4} \pi ^{2}}-\frac {147 x \sin \left (b^{2} \pi \,x^{2}\right )}{16 b^{7} \pi ^{4}}+\frac {x^{5} \sin \left (b^{2} \pi \,x^{2}\right )}{4 b^{3} \pi ^{2}}+\frac {531 \,\mathrm {S}\left (b x \sqrt {2}\right ) \sqrt {2}}{32 b^{8} \pi ^{4}} \]
command
integrate(x^7*fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {16 \, \pi ^{3} b^{8} x^{7} + 952 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 1372 \, \pi b^{4} x^{3} - 224 \, {\left (\pi ^{3} b^{7} x^{6} - 24 \, \pi b^{3} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + 3717 \, \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 28 \, {\left ({\left (4 \, \pi ^{2} b^{6} x^{5} - 147 \, b^{2} x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 48 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{224 \, \pi ^{4} b^{9}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{7} {\rm fresnelc}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ), x\right ) \]