\(\int x^8 \cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelS}(b x) \, dx\) [91]

Optimal result

Integrand size = 20, antiderivative size = 307 \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\frac {35 x^4}{8 b^5 \pi ^3}-\frac {x^8}{16 b \pi }-\frac {40 \cos \left (b^2 \pi x^2\right )}{b^9 \pi ^5}+\frac {5 x^4 \cos \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}+\frac {105 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^9 \pi ^4}-\frac {105 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}+\frac {105 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {35 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {55 x^2 \sin \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \] Output:

35/8*x^4/b^5/Pi^3-1/16*x^8/b/Pi-40*cos(b^2*Pi*x^2)/b^9/Pi^5+5/2*x^4*cos(b^ 
2*Pi*x^2)/b^5/Pi^3-105*x*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b^8/Pi^4+7*x^5* 
cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b^4/Pi^2+105/2*FresnelC(b*x)*FresnelS(b* 
x)/b^9/Pi^4-105/8*I*x^2*hypergeom([1, 1],[3/2, 2],-1/2*I*b^2*Pi*x^2)/b^7/P 
i^4+105/8*I*x^2*hypergeom([1, 1],[3/2, 2],1/2*I*b^2*Pi*x^2)/b^7/Pi^4-35*x^ 
3*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^6/Pi^3+x^7*FresnelS(b*x)*sin(1/2*b^2 
*Pi*x^2)/b^2/Pi-55/4*x^2*sin(b^2*Pi*x^2)/b^7/Pi^4+1/4*x^6*sin(b^2*Pi*x^2)/ 
b^3/Pi^2
 

Mathematica [F]

\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx \] Input:

Integrate[x^8*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x],x]
 

Output:

Integrate[x^8*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x], x]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^8 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right ) \, dx\)

\(\Big \downarrow \) 7016

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^7 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3860

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^6 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^6 \sin \left (\frac {1}{2} b^2 \pi x^2\right )^2dx^2}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3790

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {\int x^6dx^2}{2}-\frac {1}{2} \int x^6 \cos \left (b^2 \pi x^2\right )dx^2}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^8}{8}-\frac {1}{2} \int x^6 \cos \left (b^2 \pi x^2\right )dx^2}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^8}{8}-\frac {1}{2} \int x^6 \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (-\frac {3 \int -x^4 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \int x^4 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \int x^4 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \int x^2 \cos \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \int x^2 \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {\int -\sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}+\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3118

\(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 7008

\(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^5 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 3860

\(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^4 \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^4 \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {2 \int x^2 \cos \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {2 \int x^2 \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {2 \left (\frac {\int -\sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}+\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 3118

\(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 7016

\(\displaystyle -\frac {7 \left (\frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^3 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 3860

\(\displaystyle -\frac {7 \left (\frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^2 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 \left (\frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )^2dx^2}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 3790

\(\displaystyle -\frac {7 \left (\frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {\int x^2dx^2}{2}-\frac {1}{2} \int x^2 \cos \left (b^2 \pi x^2\right )dx^2}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {7 \left (\frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^4}{4}-\frac {1}{2} \int x^2 \cos \left (b^2 \pi x^2\right )dx^2}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 \left (\frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^4}{4}-\frac {1}{2} \int x^2 \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^8}{8}}{2 \pi b}\)

Input:

Int[x^8*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x],x]
 

Output:

$Aborted
 

Maple [F]

\[\int x^{8} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelS}\left (b x \right )d x\]

Input:

int(x^8*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)
 

Output:

int(x^8*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)
 

Fricas [F]

\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \] Input:

integrate(x^8*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

integral(x^8*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)
 

Sympy [F]

\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^{8} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \] Input:

integrate(x**8*cos(1/2*b**2*pi*x**2)*fresnels(b*x),x)
 

Output:

Integral(x**8*cos(pi*b**2*x**2/2)*fresnels(b*x), x)
 

Maxima [F]

\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \] Input:

integrate(x^8*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="maxima")
 

Output:

integrate(x^8*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)
 

Giac [F]

\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \] Input:

integrate(x^8*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="giac")
 

Output:

integrate(x^8*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^8\,\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \] Input:

int(x^8*FresnelS(b*x)*cos((Pi*b^2*x^2)/2),x)
 

Output:

int(x^8*FresnelS(b*x)*cos((Pi*b^2*x^2)/2), x)
 

Reduce [F]

\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^{8} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {FresnelS}\left (b x \right )d x \] Input:

int(x^8*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)
 

Output:

int(x^8*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)