\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x) \, dx \]
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)*fresnelc(b*x),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {x^{7} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi b^{2}} - \frac {5 x^{6} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{6 \pi ^{2} b^{3}} - \frac {x^{6} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{3 \pi ^{2} b^{3}} + \frac {7 x^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{2} b^{4}} - \frac {5 x^{4} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{3} b^{5}} - \frac {35 x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{3} b^{6}} + \frac {40 x^{2} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{4} b^{7}} + \frac {25 x^{2} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{4} b^{7}} - \frac {105 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{4} b^{8}} + \frac {80 \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{5} b^{9}} + \frac {105 C^{2}\left (b x\right )}{2 \pi ^{4} b^{9}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________