110.89 Problem number 136

\[ \int x \text {FresnelC}(a+b x) \, dx \]

Optimal antiderivative \[ -\frac {a^{2} \FresnelC \left (b x +a \right )}{2 b^{2}}+\frac {x^{2} \FresnelC \left (b x +a \right )}{2}+\frac {\mathrm {S}\left (b x +a \right )}{2 b^{2} \pi }+\frac {a \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{b^{2} \pi }-\frac {\left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{2 b^{2} \pi } \]

command

integrate(x*fresnel_cos(b*x+a),x, algorithm="fricas")

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

\[ \frac {\pi b^{3} x^{2} \operatorname {C}\left (b x + a\right ) - \pi a^{2} \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - {\left (b^{2} x - a b\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) + \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right )}{2 \, \pi b^{3}} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (x {\rm fresnelc}\left (b x + a\right ), x\right ) \]