110.100 Problem number 163

\[ \int x^2 \text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Optimal antiderivative \[ \left (\frac {1}{12}+\frac {i}{12}\right ) {\mathrm e}^{-\frac {3 a}{b n}+\frac {9 i}{2 b^{2} d^{2} n^{2} \pi }} x^{3} \erf \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {3}{n}+i a b \,d^{2} \pi +i b^{2} d^{2} \pi \ln \left (c \,x^{n}\right )\right )}{b d \sqrt {\pi }}\right ) \left (c \,x^{n}\right )^{-\frac {3}{n}}+\left (-\frac {1}{12}-\frac {i}{12}\right ) {\mathrm e}^{-\frac {3 a}{b n}-\frac {9 i}{2 b^{2} d^{2} n^{2} \pi }} x^{3} \erfi \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {3}{n}-i a b \,d^{2} \pi -i b^{2} d^{2} \pi \ln \left (c \,x^{n}\right )\right )}{b d \sqrt {\pi }}\right ) \left (c \,x^{n}\right )^{-\frac {3}{n}}+\frac {x^{3} \FresnelC \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3} \]

command

integrate(x^2*fresnel_cos(d*(a+b*log(c*x^n))),x, algorithm="fricas")

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

\[ \frac {1}{3} \, x^{3} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \frac {1}{6} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} - \frac {9 i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{6} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} + \frac {9 i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \frac {1}{6} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} - \frac {9 i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{6} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} + \frac {9 i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (x^{2} {\rm fresnelc}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]