110.13 Problem number 14

\[ \int \frac {S(b x)}{x^6} \, dx \]

Optimal antiderivative \[ -\frac {b^{3} \pi \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{40 x^{2}}-\frac {\mathrm {S}\left (b x \right )}{5 x^{5}}-\frac {b^{5} \pi ^{2} \sinIntegral \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{80}-\frac {b \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{20 x^{4}} \]

command

integrate(fresnel_sin(b*x)/x^6,x, algorithm="fricas")

Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output

\[ -\frac {\pi ^{2} b^{5} x^{5} \operatorname {Si}\left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 2 \, \pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 4 \, b x \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 16 \, \operatorname {S}\left (b x\right )}{80 \, x^{5}} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right )}{x^{6}}, x\right ) \]