\[ \int x^2 \text {CosIntegral}(b x) \sin (b x) \, dx \]
Optimal antiderivative \[ \frac {x^{2}}{4 b}-\frac {\cosineIntegral \left (2 b x \right )}{b^{3}}+\frac {2 \cosineIntegral \left (b x \right ) \cos \left (b x \right )}{b^{3}}-\frac {x^{2} \cosineIntegral \left (b x \right ) \cos \left (b x \right )}{b}+\frac {\cos ^{2}\left (b x \right )}{4 b^{3}}-\frac {\ln \left (x \right )}{b^{3}}+\frac {2 x \cosineIntegral \left (b x \right ) \sin \left (b x \right )}{b^{2}}+\frac {x \cos \left (b x \right ) \sin \left (b x \right )}{2 b^{2}}-\frac {\sin ^{2}\left (b x \right )}{b^{3}} \]
command
integrate(x^2*fresnel_cos(b*x)*sin(b*x),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\pi ^{2} b^{3} x^{2} - 2 \, \pi ^{2} b\right )} \cos \left (b x\right ) \operatorname {C}\left (b x\right ) + \sqrt {b^{2}} {\left ({\left (2 \, \pi ^{2} - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right ) + \pi \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} {\left ({\left (2 \, \pi ^{2} - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right ) + \pi \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) - {\left (2 \, \pi ^{2} - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) - {\left (2 \, \pi ^{2} - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - 2 \, {\left (\pi b^{2} x \cos \left (b x\right ) - 2 \, \pi b \sin \left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 2 \, {\left (2 \, \pi ^{2} b^{2} x \operatorname {C}\left (b x\right ) - b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (b x\right )}{2 \, \pi ^{2} b^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{2} \operatorname {Ci}\left (b x\right ) \sin \left (b x\right ), x\right ) \]