\[ \int \frac {\text {CosIntegral}(a+b x)}{x^2} \, dx \]
Optimal antiderivative \[ -\frac {b \cosineIntegral \! \left (b x +a \right )}{a}-\frac {\cosineIntegral \! \left (b x +a \right )}{x}+\frac {b \cosineIntegral \! \left (b x \right ) \cos \! \left (a \right )}{a}-\frac {b \sinIntegral \! \left (b x \right ) \sin \! \left (a \right )}{a} \]
command
integrate(fresnel_cos(b*x+a)/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
\[ -\frac {{\left (\Re \left ( \operatorname {Ci}\left (b x + a\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} + \Re \left ( \operatorname {Ci}\left (b x\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} + \Re \left ( \operatorname {Ci}\left (-b x - a\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} + \Re \left ( \operatorname {Ci}\left (-b x\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, \Im \left ( \operatorname {Ci}\left (b x\right ) \right ) \tan \left (\frac {1}{2} \, a\right ) - 2 \, \Im \left ( \operatorname {Ci}\left (-b x\right ) \right ) \tan \left (\frac {1}{2} \, a\right ) + 4 \, \operatorname {Si}\left (b x\right ) \tan \left (\frac {1}{2} \, a\right ) + \Re \left ( \operatorname {Ci}\left (b x + a\right ) \right ) - \Re \left ( \operatorname {Ci}\left (b x\right ) \right ) + \Re \left ( \operatorname {Ci}\left (-b x - a\right ) \right ) - \Re \left ( \operatorname {Ci}\left (-b x\right ) \right )\right )} b}{2 \, {\left (a \tan \left (\frac {1}{2} \, a\right )^{2} + a\right )}} - \frac {\operatorname {Ci}\left (b x + a\right )}{x} \]