41.9 Problem number 80

\[ \int x \text {CosIntegral}(b x)^2 \, dx \]

Optimal antiderivative \[ \frac {x^{2} \cosineIntegral \! \left (b x \right )^{2}}{2}+\frac {\cosineIntegral \! \left (2 b x \right )}{2 b^{2}}-\frac {\cosineIntegral \! \left (b x \right ) \cos \! \left (b x \right )}{b^{2}}+\frac {\ln \! \left (x \right )}{2 b^{2}}-\frac {x \cosineIntegral \! \left (b x \right ) \sin \! \left (b x \right )}{b}+\frac {\sin ^{2}\left (b x \right )}{2 b^{2}} \]

command

integrate(x*fresnel_cos(b*x)^2,x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \text {could not integrate} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \frac {1}{2} \, x^{2} \operatorname {Ci}\left (b x\right )^{2} - {\left (\frac {x \sin \left (b x\right )}{b} + \frac {\cos \left (b x\right )}{b^{2}}\right )} \operatorname {Ci}\left (b x\right ) - \frac {\cos \left (2 \, b x\right ) - \operatorname {Ci}\left (2 \, b x\right ) - \operatorname {Ci}\left (-2 \, b x\right ) - 2 \, \log \left (x\right )}{4 \, b^{2}} \]