Optimal antiderivative \[ -\frac {35 x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 b^{5} \pi ^{3}}+\frac {x^{7} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 b \pi }-\frac {105 \,\mathrm {S}\left (b x \right )}{8 b^{8} \pi ^{4}}+\frac {x^{8} \mathrm {S}\left (b x \right )}{8}+\frac {105 x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 b^{7} \pi ^{4}}-\frac {7 x^{5} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 b^{3} \pi ^{2}} \]
command
integrate(x^7*fresnel_sin(b*x),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{8} \, x^{8} \operatorname {S}\left (b x\right ) - \frac {\sqrt {\frac {1}{2}} {\left (\left (105 i + 105\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) - \left (105 i - 105\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right ) - 4 \, {\left (\sqrt {\frac {1}{2}} \pi ^{4} b^{7} x^{7} - 35 \, \sqrt {\frac {1}{2}} \pi ^{2} b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 28 \, {\left (\sqrt {\frac {1}{2}} \pi ^{3} b^{5} x^{5} - 15 \, \sqrt {\frac {1}{2}} \pi b x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )}}{16 \, \pi ^{5} b^{8}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int x^{7} {\rm fresnels}\left (b x\right )\,{d x} \]________________________________________________________________________________________