\(\int x^7 \cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelS}(b x) \, dx\) [92]

Optimal result

Integrand size = 20, antiderivative size = 217 \[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\frac {4 x^3}{b^5 \pi ^3}-\frac {x^7}{14 b \pi }+\frac {17 x^3 \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^8 \pi ^4}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}+\frac {531 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}-\frac {24 x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {147 x \sin \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}+\frac {x^5 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \] Output:

4*x^3/b^5/Pi^3-1/14*x^7/b/Pi+17/8*x^3*cos(b^2*Pi*x^2)/b^5/Pi^3-48*cos(1/2* 
b^2*Pi*x^2)*FresnelS(b*x)/b^8/Pi^4+6*x^4*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x) 
/b^4/Pi^2+531/32*FresnelS(2^(1/2)*b*x)*2^(1/2)/b^8/Pi^4-24*x^2*FresnelS(b* 
x)*sin(1/2*b^2*Pi*x^2)/b^6/Pi^3+x^6*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^2/ 
Pi-147/16*x*sin(b^2*Pi*x^2)/b^7/Pi^4+1/4*x^5*sin(b^2*Pi*x^2)/b^3/Pi^2
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.75 \[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\frac {896 b^3 \pi x^3-16 b^7 \pi ^3 x^7+476 b^3 \pi x^3 \cos \left (b^2 \pi x^2\right )+3717 \sqrt {2} \operatorname {FresnelS}\left (\sqrt {2} b x\right )+224 \operatorname {FresnelS}(b x) \left (6 \left (-8+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )+b^2 \pi x^2 \left (-24+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )-2058 b x \sin \left (b^2 \pi x^2\right )+56 b^5 \pi ^2 x^5 \sin \left (b^2 \pi x^2\right )}{224 b^8 \pi ^4} \] Input:

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

Output:

(896*b^3*Pi*x^3 - 16*b^7*Pi^3*x^7 + 476*b^3*Pi*x^3*Cos[b^2*Pi*x^2] + 3717* 
Sqrt[2]*FresnelS[Sqrt[2]*b*x] + 224*FresnelS[b*x]*(6*(-8 + b^4*Pi^2*x^4)*C 
os[(b^2*Pi*x^2)/2] + b^2*Pi*x^2*(-24 + b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2]) 
- 2058*b*x*Sin[b^2*Pi*x^2] + 56*b^5*Pi^2*x^5*Sin[b^2*Pi*x^2])/(224*b^8*Pi^ 
4)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(439\) vs. \(2(217)=434\).

Time = 2.08 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.02, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {7016, 3872, 15, 3867, 3866, 3867, 3832, 7008, 3866, 3867, 3832, 7016, 3872, 15, 3867, 3832, 7006, 3832}

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.

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

\(\Big \downarrow \) 7016

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

\(\Big \downarrow \) 3872

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

\(\Big \downarrow \) 15

\(\displaystyle -\frac {6 \int x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^7}{14}-\frac {1}{2} \int x^6 \cos \left (b^2 \pi x^2\right )dx}{\pi b}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3867

\(\displaystyle -\frac {6 \int x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \int x^4 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3866

\(\displaystyle -\frac {6 \int x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \int x^2 \cos \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3867

\(\displaystyle -\frac {6 \int x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3832

\(\displaystyle -\frac {6 \int x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}\)

\(\Big \downarrow \) 7008

\(\displaystyle -\frac {6 \left (\frac {4 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^4 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}\)

\(\Big \downarrow \) 3866

\(\displaystyle -\frac {6 \left (\frac {4 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {3 \int x^2 \cos \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}\)

\(\Big \downarrow \) 3867

\(\displaystyle -\frac {6 \left (\frac {4 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}\)

\(\Big \downarrow \) 3832

\(\displaystyle -\frac {6 \left (\frac {4 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}\)

\(\Big \downarrow \) 7016

\(\displaystyle -\frac {6 \left (\frac {4 \left (-\frac {2 \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^2 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}\)

\(\Big \downarrow \) 3872

\(\displaystyle -\frac {6 \left (\frac {4 \left (-\frac {2 \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {\int x^2dx}{2}-\frac {1}{2} \int x^2 \cos \left (b^2 \pi x^2\right )dx}{\pi b}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {6 \left (\frac {4 \left (-\frac {2 \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^3}{6}-\frac {1}{2} \int x^2 \cos \left (b^2 \pi x^2\right )dx}{\pi b}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}\)

\(\Big \downarrow \) 3867

\(\displaystyle -\frac {6 \left (\frac {4 \left (-\frac {2 \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^3}{6}}{\pi b}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}\)

\(\Big \downarrow \) 3832

\(\displaystyle -\frac {6 \left (\frac {4 \left (-\frac {2 \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^3}{6}}{\pi b}\right )}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}\)

\(\Big \downarrow \) 7006

\(\displaystyle -\frac {6 \left (\frac {4 \left (-\frac {2 \left (\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^3}{6}}{\pi b}\right )}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {6 \left (-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}+\frac {4 \left (\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^2}-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^3}{6}}{\pi b}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^5 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^7}{14}}{\pi b}\)

Input:

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

Output:

(x^6*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) - (6*(-((x^4*Cos[(b^2*Pi* 
x^2)/2]*FresnelS[b*x])/(b^2*Pi)) + (-1/2*(x^3*Cos[b^2*Pi*x^2])/(b^2*Pi) + 
(3*(-1/2*FresnelS[Sqrt[2]*b*x]/(Sqrt[2]*b^3*Pi) + (x*Sin[b^2*Pi*x^2])/(2*b 
^2*Pi)))/(2*b^2*Pi))/(2*b*Pi) + (4*((-2*(-((Cos[(b^2*Pi*x^2)/2]*FresnelS[b 
*x])/(b^2*Pi)) + FresnelS[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2*Pi)))/(b^2*Pi) + (x^ 
2*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) - (x^3/6 + (FresnelS[Sqrt[2] 
*b*x]/(2*Sqrt[2]*b^3*Pi) - (x*Sin[b^2*Pi*x^2])/(2*b^2*Pi))/2)/(b*Pi)))/(b^ 
2*Pi)))/(b^2*Pi) - (x^7/14 + (-1/2*(x^5*Sin[b^2*Pi*x^2])/(b^2*Pi) + (5*(-1 
/2*(x^3*Cos[b^2*Pi*x^2])/(b^2*Pi) + (3*(-1/2*FresnelS[Sqrt[2]*b*x]/(Sqrt[2 
]*b^3*Pi) + (x*Sin[b^2*Pi*x^2])/(2*b^2*Pi)))/(2*b^2*Pi)))/(2*b^2*Pi))/2)/( 
b*Pi)
 

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 3832

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

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 7006

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

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 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 = 8.76 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.48

method	result	size
default	\(\frac {\frac {\operatorname {FresnelS}\left (b x \right ) \left (\frac {b^{6} x^{6} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {6 \left (-\frac {b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}}{\pi }\right )}{\pi }\right )}{b^{7}}-\frac {\frac {\frac {1}{7} \pi ^{2} b^{7} x^{7}-8 b^{3} x^{3}}{2 \pi ^{3}}+\frac {-\frac {3 \pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {9 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{4 \pi }\right )}{2}-12 \sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{\pi ^{4}}-\frac {\frac {\pi \,b^{5} x^{5} \sin \left (b^{2} \pi \,x^{2}\right )}{2}-\frac {5 \pi \left (-\frac {b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\frac {3 b x \sin \left (b^{2} \pi \,x^{2}\right )}{4 \pi }-\frac {3 \sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{8 \pi }}{\pi }\right )}{2}-\frac {12 b x \sin \left (b^{2} \pi \,x^{2}\right )}{\pi }+\frac {6 \sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{\pi }}{2 \pi ^{3}}}{b^{7}}}{b}\)	\(321\)

Input:

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

Output:

(FresnelS(b*x)/b^7*(1/Pi*b^6*x^6*sin(1/2*b^2*Pi*x^2)-6/Pi*(-1/Pi*b^4*x^4*c 
os(1/2*b^2*Pi*x^2)+4/Pi*(1/Pi*b^2*x^2*sin(1/2*b^2*Pi*x^2)+2/Pi^2*cos(1/2*b 
^2*Pi*x^2))))-1/b^7*(1/2/Pi^3*(1/7*Pi^2*b^7*x^7-8*b^3*x^3)+3/Pi^4*(-1/2*Pi 
*b^3*x^3*cos(b^2*Pi*x^2)+3/2*Pi*(1/2/Pi*b*x*sin(b^2*Pi*x^2)-1/4/Pi*2^(1/2) 
*FresnelS(2^(1/2)*b*x))-4*2^(1/2)*FresnelS(2^(1/2)*b*x))-1/2/Pi^3*(1/2*Pi* 
b^5*x^5*sin(b^2*Pi*x^2)-5/2*Pi*(-1/2/Pi*b^3*x^3*cos(b^2*Pi*x^2)+3/2/Pi*(1/ 
2/Pi*b*x*sin(b^2*Pi*x^2)-1/4/Pi*2^(1/2)*FresnelS(2^(1/2)*b*x)))-12/Pi*b*x* 
sin(b^2*Pi*x^2)+6/Pi*2^(1/2)*FresnelS(2^(1/2)*b*x))))/b
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.78 \[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=-\frac {16 \, \pi ^{3} b^{8} x^{7} - 952 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 420 \, \pi b^{4} x^{3} - 1344 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - 3717 \, \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 28 \, {\left ({\left (4 \, \pi ^{2} b^{6} x^{5} - 147 \, b^{2} x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 8 \, {\left (\pi ^{3} b^{7} x^{6} - 24 \, \pi b^{3} x^{2}\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{224 \, \pi ^{4} b^{9}} \] Input:

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

Output:

-1/224*(16*pi^3*b^8*x^7 - 952*pi*b^4*x^3*cos(1/2*pi*b^2*x^2)^2 - 420*pi*b^ 
4*x^3 - 1344*(pi^2*b^5*x^4 - 8*b)*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x) - 3 
717*sqrt(2)*sqrt(b^2)*fresnel_sin(sqrt(2)*sqrt(b^2)*x) - 28*((4*pi^2*b^6*x 
^5 - 147*b^2*x)*cos(1/2*pi*b^2*x^2) + 8*(pi^3*b^7*x^6 - 24*pi*b^3*x^2)*fre 
snel_sin(b*x))*sin(1/2*pi*b^2*x^2))/(pi^4*b^9)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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