\(\int x^8 \operatorname {FresnelS}(b x) \sin (\frac {1}{2} b^2 \pi x^2) \, dx\) [71]

Optimal result

Integrand size = 20, antiderivative size = 232 \[ \int x^8 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}+\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^6 \pi ^3}-\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^2 \pi }+\frac {105 \operatorname {FresnelS}(b x)^2}{2 b^9 \pi ^4}-\frac {105 x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac {7 x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {40 \sin \left (b^2 \pi x^2\right )}{b^9 \pi ^5}+\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3} \] Output:

105/4*x^2/b^7/Pi^4-7/12*x^6/b^3/Pi^2+55/4*x^2*cos(b^2*Pi*x^2)/b^7/Pi^4-1/4 
*x^6*cos(b^2*Pi*x^2)/b^3/Pi^2+35*x^3*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b^6 
/Pi^3-x^7*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b^2/Pi+105/2*FresnelS(b*x)^2/b 
^9/Pi^4-105*x*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^8/Pi^4+7*x^5*FresnelS(b* 
x)*sin(1/2*b^2*Pi*x^2)/b^4/Pi^2-40*sin(b^2*Pi*x^2)/b^9/Pi^5+5/2*x^4*sin(b^ 
2*Pi*x^2)/b^5/Pi^3
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00 \[ \int x^8 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}+\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^6 \pi ^3}-\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^2 \pi }+\frac {105 \operatorname {FresnelS}(b x)^2}{2 b^9 \pi ^4}-\frac {105 x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac {7 x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {40 \sin \left (b^2 \pi x^2\right )}{b^9 \pi ^5}+\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3} \] Input:

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

Output:

(105*x^2)/(4*b^7*Pi^4) - (7*x^6)/(12*b^3*Pi^2) + (55*x^2*Cos[b^2*Pi*x^2])/ 
(4*b^7*Pi^4) - (x^6*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (35*x^3*Cos[(b^2*Pi*x^ 
2)/2]*FresnelS[b*x])/(b^6*Pi^3) - (x^7*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/ 
(b^2*Pi) + (105*FresnelS[b*x]^2)/(2*b^9*Pi^4) - (105*x*FresnelS[b*x]*Sin[( 
b^2*Pi*x^2)/2])/(b^8*Pi^4) + (7*x^5*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^ 
4*Pi^2) - (40*Sin[b^2*Pi*x^2])/(b^9*Pi^5) + (5*x^4*Sin[b^2*Pi*x^2])/(2*b^5 
*Pi^3)
 

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.

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

\(\Big \downarrow \) 7008

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

\(\Big \downarrow \) 3860

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 7016

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

\(\Big \downarrow \) 3860

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3790

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 7008

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

\(\Big \downarrow \) 3860

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 7016

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.73 \[ \int x^8 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=-\frac {2 \, \pi ^{3} b^{6} x^{6} - 75 \, \pi b^{2} x^{2} + 3 \, {\left (\pi ^{3} b^{6} x^{6} - 55 \, \pi b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 6 \, {\left (\pi ^{4} b^{7} x^{7} - 35 \, \pi ^{2} b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - 315 \, \pi \operatorname {S}\left (b x\right )^{2} - 6 \, {\left (5 \, {\left (\pi ^{2} b^{4} x^{4} - 16\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 7 \, {\left (\pi ^{3} b^{5} x^{5} - 15 \, \pi b x\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{6 \, \pi ^{5} b^{9}} \] Input:

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

Output:

-1/6*(2*pi^3*b^6*x^6 - 75*pi*b^2*x^2 + 3*(pi^3*b^6*x^6 - 55*pi*b^2*x^2)*co 
s(1/2*pi*b^2*x^2)^2 + 6*(pi^4*b^7*x^7 - 35*pi^2*b^3*x^3)*cos(1/2*pi*b^2*x^ 
2)*fresnel_sin(b*x) - 315*pi*fresnel_sin(b*x)^2 - 6*(5*(pi^2*b^4*x^4 - 16) 
*cos(1/2*pi*b^2*x^2) + 7*(pi^3*b^5*x^5 - 15*pi*b*x)*fresnel_sin(b*x))*sin( 
1/2*pi*b^2*x^2))/(pi^5*b^9)
 

Sympy [A] (verification not implemented)

Time = 13.81 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.30 \[ \int x^8 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\begin {cases} - \frac {x^{7} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{\pi b^{2}} - \frac {x^{6} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{3 \pi ^{2} b^{3}} - \frac {5 x^{6} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{6 \pi ^{2} b^{3}} + \frac {7 x^{5} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{\pi ^{2} b^{4}} + \frac {5 x^{4} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{3} b^{5}} + \frac {35 x^{3} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{\pi ^{3} b^{6}} + \frac {25 x^{2} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{4} b^{7}} + \frac {40 x^{2} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{4} b^{7}} - \frac {105 x \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{\pi ^{4} b^{8}} - \frac {80 \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{5} b^{9}} + \frac {105 S^{2}\left (b x\right )}{2 \pi ^{4} b^{9}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-x**7*cos(pi*b**2*x**2/2)*fresnels(b*x)/(pi*b**2) - x**6*sin(pi 
*b**2*x**2/2)**2/(3*pi**2*b**3) - 5*x**6*cos(pi*b**2*x**2/2)**2/(6*pi**2*b 
**3) + 7*x**5*sin(pi*b**2*x**2/2)*fresnels(b*x)/(pi**2*b**4) + 5*x**4*sin( 
pi*b**2*x**2/2)*cos(pi*b**2*x**2/2)/(pi**3*b**5) + 35*x**3*cos(pi*b**2*x** 
2/2)*fresnels(b*x)/(pi**3*b**6) + 25*x**2*sin(pi*b**2*x**2/2)**2/(2*pi**4* 
b**7) + 40*x**2*cos(pi*b**2*x**2/2)**2/(pi**4*b**7) - 105*x*sin(pi*b**2*x* 
*2/2)*fresnels(b*x)/(pi**4*b**8) - 80*sin(pi*b**2*x**2/2)*cos(pi*b**2*x**2 
/2)/(pi**5*b**9) + 105*fresnels(b*x)**2/(2*pi**4*b**9), Ne(b, 0)), (0, Tru 
e))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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