\[ \int \frac {\text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx \]
Optimal antiderivative \[ -\frac {b^{3} \pi }{756 x^{6}}+\frac {b^{7} \pi ^{3}}{3780 x^{2}}-\frac {11 b^{3} \pi \cos \left (b^{2} \pi \,x^{2}\right )}{3024 x^{6}}+\frac {5 b^{7} \pi ^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2016 x^{2}}-\frac {b^{2} \pi \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{63 x^{7}}+\frac {b^{6} \pi ^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{945 x^{3}}+\frac {b^{9} \pi ^{5} \FresnelC \left (b x \right )^{2}}{1890}+\frac {83 b^{9} \pi ^{4} \sinIntegral \left (b^{2} \pi \,x^{2}\right )}{30240}-\frac {\FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{9 x^{9}}+\frac {b^{4} \pi ^{2} \FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{315 x^{5}}-\frac {b^{8} \pi ^{4} \FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{945 x}-\frac {b \sin \left (b^{2} \pi \,x^{2}\right )}{144 x^{8}}+\frac {67 b^{5} \pi ^{2} \sin \left (b^{2} \pi \,x^{2}\right )}{30240 x^{4}} \]
command
integrate(fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2)/x^10,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {16 \, \pi ^{5} b^{9} x^{9} \operatorname {C}\left (b x\right )^{2} + 83 \, \pi ^{4} b^{9} x^{9} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 67 \, \pi ^{3} b^{7} x^{7} + 70 \, \pi b^{3} x^{3} + 10 \, {\left (15 \, \pi ^{3} b^{7} x^{7} - 22 \, \pi b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 32 \, {\left (\pi ^{3} b^{6} x^{6} - 15 \, \pi b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + 2 \, {\left ({\left (67 \, \pi ^{2} b^{5} x^{5} - 210 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 16 \, {\left (\pi ^{4} b^{8} x^{8} - 3 \, \pi ^{2} b^{4} x^{4} + 105\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{30240 \, x^{9}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\rm fresnelc}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}}, x\right ) \]