Optimal antiderivative \[ -\frac {105 x^{2}}{16 b^{6} \pi ^{4}}+\frac {7 x^{6}}{48 b^{2} \pi ^{2}}-\frac {55 x^{2} \cos \left (b^{2} \pi \,x^{2}\right )}{16 b^{6} \pi ^{4}}+\frac {x^{6} \cos \left (b^{2} \pi \,x^{2}\right )}{16 b^{2} \pi ^{2}}-\frac {35 x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{4 b^{5} \pi ^{3}}+\frac {x^{7} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{4 b \pi }-\frac {105 \mathrm {S}\left (b x \right )^{2}}{8 b^{8} \pi ^{4}}+\frac {x^{8} \mathrm {S}\left (b x \right )^{2}}{8}+\frac {105 x \,\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 b^{7} \pi ^{4}}-\frac {7 x^{5} \mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 b^{3} \pi ^{2}}+\frac {10 \sin \left (b^{2} \pi \,x^{2}\right )}{b^{8} \pi ^{5}}-\frac {5 x^{4} \sin \left (b^{2} \pi \,x^{2}\right )}{8 b^{4} \pi ^{3}} \]
command
integrate(x^7*fresnel_sin(b*x)^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, \pi ^{3} b^{6} x^{6} - 75 \, \pi b^{2} x^{2} + 3 \, {\left (\pi ^{3} b^{6} x^{6} - 55 \, \pi b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 6 \, {\left (\pi ^{4} b^{7} x^{7} - 35 \, \pi ^{2} b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - 3 \, {\left (105 \, \pi - \pi ^{5} b^{8} x^{8}\right )} \operatorname {S}\left (b x\right )^{2} - 6 \, {\left (5 \, {\left (\pi ^{2} b^{4} x^{4} - 16\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 7 \, {\left (\pi ^{3} b^{5} x^{5} - 15 \, \pi b x\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{24 \, \pi ^{5} b^{8}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{7} {\rm fresnels}\left (b x\right )^{2}, x\right ) \]