41.5 Problem number 75

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

Optimal antiderivative \[ -\frac {\cosineIntegral \! \left (b x \right )}{x}-\frac {\cos \! \left (b x \right )}{x}-b \sinIntegral \! \left (b x \right ) \]

command

integrate(fresnel_cos(b*x)/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

\[ -\frac {b x \Im \left ( \operatorname {Ci}\left (b x\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} - b x \Im \left ( \operatorname {Ci}\left (-b x\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} + 2 \, b x \operatorname {Si}\left (b x\right ) \tan \left (\frac {1}{2} \, b x\right )^{2} + b x \Im \left ( \operatorname {Ci}\left (b x\right ) \right ) - b x \Im \left ( \operatorname {Ci}\left (-b x\right ) \right ) + 2 \, b x \operatorname {Si}\left (b x\right ) - 2 \, \tan \left (\frac {1}{2} \, b x\right )^{2} + 2}{2 \, {\left (x \tan \left (\frac {1}{2} \, b x\right )^{2} + x\right )}} - \frac {\operatorname {Ci}\left (b x\right )}{x} \]