110.123 Problem number 202

\[ \int x^6 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \]

Optimal antiderivative \[ -\frac {15 x^{2}}{4 b^{5} \pi ^{3}}+\frac {x^{6}}{12 b \pi }+\frac {7 x^{2} \cos \left (b^{2} \pi \,x^{2}\right )}{4 b^{5} \pi ^{3}}+\frac {15 x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{b^{6} \pi ^{3}}-\frac {x^{5} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{b^{2} \pi }-\frac {15 \FresnelC \left (b x \right )^{2}}{2 b^{7} \pi ^{3}}+\frac {5 x^{3} \FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{4} \pi ^{2}}-\frac {11 \sin \left (b^{2} \pi \,x^{2}\right )}{2 b^{7} \pi ^{4}}+\frac {x^{4} \sin \left (b^{2} \pi \,x^{2}\right )}{4 b^{3} \pi ^{2}} \]

command

integrate(x^6*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

\[ \frac {\pi ^{3} b^{6} x^{6} + 42 \, \pi b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 66 \, \pi b^{2} x^{2} - 12 \, {\left (\pi ^{3} b^{5} x^{5} - 15 \, \pi b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 90 \, \pi \operatorname {C}\left (b x\right )^{2} + 6 \, {\left (10 \, \pi ^{2} b^{3} x^{3} \operatorname {C}\left (b x\right ) + {\left (\pi ^{2} b^{4} x^{4} - 22\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{12 \, \pi ^{4} b^{7}} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (x^{6} {\rm fresnelc}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ), x\right ) \]