3.2.81 \(\int x^7 \cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelC}(b x) \, dx\) [181]

3.2.81.1 Optimal result

Integrand size = 20, antiderivative size = 215 \[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\frac {24 x}{b^7 \pi ^4}-\frac {3 x^5}{5 b^3 \pi ^2}-\frac {147 x \cos \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}+\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {531 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}-\frac {24 x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3} \]

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

3.2.81.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.72 \[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\frac {2655 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )+160 \operatorname {FresnelC}(b x) \left (6 \left (-8+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )+b^2 \pi x^2 \left (-24+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+2 b x \left (5 \left (-147+4 b^4 \pi ^2 x^4\right ) \cos \left (b^2 \pi x^2\right )-2 \left (-960+24 b^4 \pi ^2 x^4+85 b^2 \pi x^2 \sin \left (b^2 \pi x^2\right )\right )\right )}{160 b^8 \pi ^4} \]

input

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

output

(2655*Sqrt[2]*FresnelC[Sqrt[2]*b*x] + 160*FresnelC[b*x]*(6*(-8 + b^4*Pi^2* 
x^4)*Cos[(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]) + 2*b*x*(5*(-147 + 4*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)
 

3.2.81.3 Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(438\) vs. \(2(215)=430\).

Time = 1.96 (sec) , antiderivative size = 438, 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 = {7009, 3866, 3867, 3866, 3833, 7017, 3873, 15, 3867, 3866, 3833, 7009, 3866, 3833, 7015, 3839, 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.

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

\(\Big \downarrow \) 7009

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

\(\Big \downarrow \) 3866

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

\(\Big \downarrow \) 3867

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

\(\Big \downarrow \) 3866

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 7017

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

\(\Big \downarrow \) 3873

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 3867

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

\(\Big \downarrow \) 3866

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 7009

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

\(\Big \downarrow \) 3866

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 7015

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

\(\Big \downarrow \) 3839

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.2.81.3.1 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 3839

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_), x_Sy 
mbol] :> Int[ExpandTrigReduce[(a + b*Cos[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 3873

Int[Cos[(a_.) + ((b_.)*(x_)^(n_))/2]^2*(x_)^(m_.), 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 7009

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

rule 7015

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

rule 7017

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

3.2.81.4 Maple [A] (verified)

Time = 7.28 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.47

method	result	size
default	\(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) \left (\frac {b^{6} x^{6} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {6 \left (-\frac {b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}}{\pi }\right )}{\pi }\right )}{b^{7}}-\frac {\frac {\frac {3}{5} b^{5} x^{5} \pi ^{2}-24 b x}{\pi ^{4}}+\frac {\frac {3 \pi \,b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2}-\frac {9 \pi \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-12 \sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{\pi ^{4}}+\frac {-\frac {\pi \,b^{5} x^{5} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {5 \pi \left (\frac {b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {3 \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2 \pi }\right )}{2}+\frac {12 b x \cos \left (b^{2} \pi \,x^{2}\right )}{\pi }-\frac {6 \sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{\pi }}{2 \pi ^{3}}}{b^{7}}}{b}\)	\(317\)

input

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

output

(FresnelC(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*(3/Pi^4*(1/5*b^5*x^5*Pi^2-8*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)*Fresn 
elC(b*x*2^(1/2)))-4*2^(1/2)*FresnelC(b*x*2^(1/2)))+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(b*x*2^(1/2))))+12/Pi*b*x*cos(b 
^2*Pi*x^2)-6/Pi*2^(1/2)*FresnelC(b*x*2^(1/2)))))/b
 

3.2.81.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.78 \[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=-\frac {136 \, \pi ^{2} b^{6} x^{5} - 5310 \, b^{2} x - 20 \, {\left (4 \, \pi ^{2} b^{6} x^{5} - 147 \, b^{2} x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 960 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 2655 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 40 \, {\left (17 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 4 \, {\left (\pi ^{3} b^{7} x^{6} - 24 \, \pi b^{3} x^{2}\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{160 \, \pi ^{4} b^{9}} \]

input

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

output

-1/160*(136*pi^2*b^6*x^5 - 5310*b^2*x - 20*(4*pi^2*b^6*x^5 - 147*b^2*x)*co 
s(1/2*pi*b^2*x^2)^2 - 960*(pi^2*b^5*x^4 - 8*b)*cos(1/2*pi*b^2*x^2)*fresnel 
_cos(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) - 4*(pi^3*b^7*x^6 - 24*pi*b^3*x^2)*fresn 
el_cos(b*x))*sin(1/2*pi*b^2*x^2))/(pi^4*b^9)
 

3.2.81.6 Sympy [F]

\[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int x^{7} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \]

input

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

output

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

3.2.81.7 Maxima [F]

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

input

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

output

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

3.2.81.8 Giac [F]

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

input

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

output

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

3.2.81.9 Mupad [F(-1)]

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

input

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

output

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