\(\int x^7 \operatorname {FresnelS}(b x) \sin (\frac {1}{2} b^2 \pi x^2) \, dx\) [72]

Optimal result

Integrand size = 20, antiderivative size = 216 \[ \int x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {24 x}{b^7 \pi ^4}-\frac {3 x^5}{5 b^3 \pi ^2}+\frac {147 x \cos \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}-\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {531 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}+\frac {24 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^6 \pi ^3}-\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^2 \pi }-\frac {48 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac {6 x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3} \] Output:

24*x/b^7/Pi^4-3/5*x^5/b^3/Pi^2+147/16*x*cos(b^2*Pi*x^2)/b^7/Pi^4-1/4*x^5*c 
os(b^2*Pi*x^2)/b^3/Pi^2-531/32*FresnelC(2^(1/2)*b*x)*2^(1/2)/b^8/Pi^4+24*x 
^2*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b^6/Pi^3-x^6*cos(1/2*b^2*Pi*x^2)*Fres 
nelS(b*x)/b^2/Pi-48*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^8/Pi^4+6*x^4*Fresn 
elS(b*x)*sin(1/2*b^2*Pi*x^2)/b^4/Pi^2+17/8*x^3*sin(b^2*Pi*x^2)/b^5/Pi^3
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.71 \[ \int x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {-2655 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )-160 \operatorname {FresnelS}(b x) \left (b^2 \pi x^2 \left (-24+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )-6 \left (-8+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+2 b x \left (\left (735-20 b^4 \pi ^2 x^4\right ) \cos \left (b^2 \pi x^2\right )+2 \left (960-24 b^4 \pi ^2 x^4+85 b^2 \pi x^2 \sin \left (b^2 \pi x^2\right )\right )\right )}{160 b^8 \pi ^4} \] Input:

Integrate[x^7*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2],x]
 

Output:

(-2655*Sqrt[2]*FresnelC[Sqrt[2]*b*x] - 160*FresnelS[b*x]*(b^2*Pi*x^2*(-24 
+ b^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2] - 6*(-8 + b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^ 
2)/2]) + 2*b*x*((735 - 20*b^4*Pi^2*x^4)*Cos[b^2*Pi*x^2] + 2*(960 - 24*b^4* 
Pi^2*x^4 + 85*b^2*Pi*x^2*Sin[b^2*Pi*x^2])))/(160*b^8*Pi^4)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(440\) vs. \(2(216)=432\).

Time = 2.11 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.04, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.850, Rules used = {7008, 3866, 3867, 3866, 3833, 7016, 3872, 15, 3867, 3866, 3833, 7008, 3866, 3833, 7014, 3838, 2009}

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.

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

\(\Big \downarrow \) 7008

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

\(\Big \downarrow \) 3866

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

\(\Big \downarrow \) 3867

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

\(\Big \downarrow \) 3866

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 7016

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

\(\Big \downarrow \) 3872

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {6 \left (-\frac {4 \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^5}{10}-\frac {1}{2} \int x^4 \cos \left (b^2 \pi x^2\right )dx}{\pi b}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {5 \left (\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\)

\(\Big \downarrow \) 3867

\(\displaystyle \frac {6 \left (-\frac {4 \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \int x^2 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {5 \left (\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\)

\(\Big \downarrow \) 3866

\(\displaystyle \frac {6 \left (-\frac {4 \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\int \cos \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {5 \left (\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {6 \left (-\frac {4 \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\right )}{\pi b^2}-\frac {x^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {5 \left (\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\)

\(\Big \downarrow \) 7008

\(\displaystyle \frac {6 \left (-\frac {4 \left (\frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^2 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\right )}{\pi b^2}-\frac {x^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {5 \left (\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\)

\(\Big \downarrow \) 3866

\(\displaystyle \frac {6 \left (-\frac {4 \left (\frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {\int \cos \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\right )}{\pi b^2}-\frac {x^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {5 \left (\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {6 \left (-\frac {4 \left (\frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\right )}{\pi b^2}-\frac {x^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {5 \left (\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\)

\(\Big \downarrow \) 7014

\(\displaystyle \frac {6 \left (-\frac {4 \left (\frac {2 \left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\int \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}\right )}{\pi b^2}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\right )}{\pi b^2}-\frac {x^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {5 \left (\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\)

\(\Big \downarrow \) 3838

\(\displaystyle \frac {6 \left (-\frac {4 \left (\frac {2 \left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\int \left (\frac {1}{2}-\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right )dx}{\pi b}\right )}{\pi b^2}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\right )}{\pi b^2}-\frac {x^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {5 \left (\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {x^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {6 \left (\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {4 \left (\frac {2 \left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {x}{2}-\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b}}{\pi b}\right )}{\pi b^2}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\right )}{\pi b^2}+\frac {\frac {5 \left (\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\)

Input:

Int[x^7*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2],x]
 

Output:

-((x^6*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^2*Pi)) + (-1/2*(x^5*Cos[b^2*P 
i*x^2])/(b^2*Pi) + (5*((-3*(-1/2*(x*Cos[b^2*Pi*x^2])/(b^2*Pi) + FresnelC[S 
qrt[2]*b*x]/(2*Sqrt[2]*b^3*Pi)))/(2*b^2*Pi) + (x^3*Sin[b^2*Pi*x^2])/(2*b^2 
*Pi)))/(2*b^2*Pi))/(2*b*Pi) + (6*((x^4*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/ 
(b^2*Pi) - (4*((-1/2*(x*Cos[b^2*Pi*x^2])/(b^2*Pi) + FresnelC[Sqrt[2]*b*x]/ 
(2*Sqrt[2]*b^3*Pi))/(2*b*Pi) - (x^2*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^ 
2*Pi) + (2*(-((x/2 - FresnelC[Sqrt[2]*b*x]/(2*Sqrt[2]*b))/(b*Pi)) + (Fresn 
elS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi)))/(b^2*Pi)))/(b^2*Pi) - (x^5/10 + ( 
(3*(-1/2*(x*Cos[b^2*Pi*x^2])/(b^2*Pi) + FresnelC[Sqrt[2]*b*x]/(2*Sqrt[2]*b 
^3*Pi)))/(2*b^2*Pi) - (x^3*Sin[b^2*Pi*x^2])/(2*b^2*Pi))/2)/(b*Pi)))/(b^2*P 
i)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3838

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_), x_Sy 
mbol] :> Int[ExpandTrigReduce[(a + b*Sin[c + d*(e + f*x)^n])^p, x], x] /; F 
reeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 1] && IGtQ[n, 1]
 

rule 3866

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(-e^ 
(n - 1))*(e*x)^(m - n + 1)*(Cos[c + d*x^n]/(d*n)), x] + Simp[e^n*((m - n + 
1)/(d*n))   Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] 
 && IGtQ[n, 0] && LtQ[n, m + 1]
 

rule 3867

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[e^(n 
 - 1)*(e*x)^(m - n + 1)*(Sin[c + d*x^n]/(d*n)), x] - Simp[e^n*((m - n + 1)/ 
(d*n))   Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && 
 IGtQ[n, 0] && LtQ[n, m + 1]
 

rule 3872

Int[(x_)^(m_.)*Sin[(a_.) + ((b_.)*(x_)^(n_))/2]^2, x_Symbol] :> Simp[1/2 
Int[x^m, x], x] - Simp[1/2   Int[x^m*Cos[2*a + b*x^n], x], x] /; FreeQ[{a, 
b, m, n}, x]
 

rule 7008

Int[FresnelS[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[(-x 
^(m - 1))*Cos[d*x^2]*(FresnelS[b*x]/(2*d)), x] + (Simp[(m - 1)/(2*d)   Int[ 
x^(m - 2)*Cos[d*x^2]*FresnelS[b*x], x], x] + Simp[1/(2*b*Pi)   Int[x^(m - 1 
)*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IG 
tQ[m, 1]
 

rule 7014

Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)]*(x_), x_Symbol] :> Simp[Sin[d*x^ 
2]*(FresnelS[b*x]/(2*d)), x] - Simp[1/(Pi*b)   Int[Sin[d*x^2]^2, x], x] /; 
FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4]
 

rule 7016

Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^( 
m - 1)*Sin[d*x^2]*(FresnelS[b*x]/(2*d)), x] + (-Simp[1/(Pi*b)   Int[x^(m - 
1)*Sin[d*x^2]^2, x], x] - Simp[(m - 1)/(2*d)   Int[x^(m - 2)*Sin[d*x^2]*Fre 
snelS[b*x], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IGtQ[m 
, 1]
 

Maple [A] (verified)

Time = 5.03 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.47

method	result	size
default	\(\frac {\frac {\operatorname {FresnelS}\left (b x \right ) \left (-\frac {b^{6} x^{6} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\frac {6 b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {24 \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{\pi }}{\pi }\right )}{b^{7}}-\frac {\frac {\frac {3}{5} \pi ^{2} b^{5} x^{5}-24 b x}{\pi ^{4}}-\frac {3 \left (\frac {\pi \,b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2}-\frac {3 \pi \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )\right )}{\pi ^{4}}-\frac {-\frac {\pi \,b^{5} x^{5} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {5 \pi \left (\frac {b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {3 \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )}{4 \pi }\right )}{2 \pi }\right )}{2}+\frac {12 b x \cos \left (b^{2} \pi \,x^{2}\right )}{\pi }-\frac {6 \sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )}{\pi }}{2 \pi ^{3}}}{b^{7}}}{b}\)	\(318\)

Input:

int(x^7*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2),x,method=_RETURNVERBOSE)
 

Output:

(FresnelS(b*x)/b^7*(-1/Pi*b^6*x^6*cos(1/2*b^2*Pi*x^2)+6/Pi*(1/Pi*b^4*x^4*s 
in(1/2*b^2*Pi*x^2)-4/Pi*(-1/Pi*b^2*x^2*cos(1/2*b^2*Pi*x^2)+2/Pi^2*sin(1/2* 
b^2*Pi*x^2))))-1/b^7*(3/Pi^4*(1/5*Pi^2*b^5*x^5-8*b*x)-3/Pi^4*(1/2*Pi*b^3*x 
^3*sin(b^2*Pi*x^2)-3/2*Pi*(-1/2/Pi*b*x*cos(b^2*Pi*x^2)+1/4/Pi*2^(1/2)*Fres 
nelC(2^(1/2)*b*x))-4*2^(1/2)*FresnelC(2^(1/2)*b*x))-1/2/Pi^3*(-1/2*Pi*b^5* 
x^5*cos(b^2*Pi*x^2)+5/2*Pi*(1/2/Pi*b^3*x^3*sin(b^2*Pi*x^2)-3/2/Pi*(-1/2/Pi 
*b*x*cos(b^2*Pi*x^2)+1/4/Pi*2^(1/2)*FresnelC(2^(1/2)*b*x)))+12/Pi*b*x*cos( 
b^2*Pi*x^2)-6/Pi*2^(1/2)*FresnelC(2^(1/2)*b*x))))/b
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.77 \[ \int x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=-\frac {56 \, \pi ^{2} b^{6} x^{5} - 2370 \, b^{2} x + 20 \, {\left (4 \, \pi ^{2} b^{6} x^{5} - 147 \, b^{2} x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 160 \, {\left (\pi ^{3} b^{7} x^{6} - 24 \, \pi b^{3} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 2655 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 40 \, {\left (17 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 24 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{160 \, \pi ^{4} b^{9}} \] Input:

integrate(x^7*fresnel_sin(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="fricas")
 

Output:

-1/160*(56*pi^2*b^6*x^5 - 2370*b^2*x + 20*(4*pi^2*b^6*x^5 - 147*b^2*x)*cos 
(1/2*pi*b^2*x^2)^2 + 160*(pi^3*b^7*x^6 - 24*pi*b^3*x^2)*cos(1/2*pi*b^2*x^2 
)*fresnel_sin(b*x) + 2655*sqrt(2)*sqrt(b^2)*fresnel_cos(sqrt(2)*sqrt(b^2)* 
x) - 40*(17*pi*b^4*x^3*cos(1/2*pi*b^2*x^2) + 24*(pi^2*b^5*x^4 - 8*b)*fresn 
el_sin(b*x))*sin(1/2*pi*b^2*x^2))/(pi^4*b^9)
 

Sympy [F]

\[ \int x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x^{7} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \] Input:

integrate(x**7*fresnels(b*x)*sin(1/2*b**2*pi*x**2),x)
 

Output:

Integral(x**7*sin(pi*b**2*x**2/2)*fresnels(b*x), x)
 

Maxima [F]

\[ \int x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x^{7} \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \] Input:

integrate(x^7*fresnel_sin(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="maxima")
 

Output:

integrate(x^7*fresnel_sin(b*x)*sin(1/2*pi*b^2*x^2), x)
 

Giac [F]

\[ \int x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x^{7} \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \] Input:

integrate(x^7*fresnel_sin(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="giac")
 

Output:

integrate(x^7*fresnel_sin(b*x)*sin(1/2*pi*b^2*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x^7\,\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \] Input:

int(x^7*FresnelS(b*x)*sin((Pi*b^2*x^2)/2),x)
 

Output:

int(x^7*FresnelS(b*x)*sin((Pi*b^2*x^2)/2), x)
 

Reduce [F]

\[ \int x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x^{7} \mathrm {FresnelS}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )d x \] Input:

int(x^7*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2),x)
 

Output:

int(x^7*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2),x)