3.2.82 \(\int x^6 \cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelC}(b x) \, dx\) [182]

3.2.82.1 Optimal result

Integrand size = 20, antiderivative size = 247 \[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=-\frac {5 x^4}{8 b^3 \pi ^2}-\frac {11 \cos \left (b^2 \pi x^2\right )}{2 b^7 \pi ^4}+\frac {x^4 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {15 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^7 \pi ^3}+\frac {15 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {15 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {15 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {7 x^2 \sin \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3} \]

output

-5/8*x^4/b^3/Pi^2-11/2*cos(b^2*Pi*x^2)/b^7/Pi^4+1/4*x^4*cos(b^2*Pi*x^2)/b^ 
3/Pi^2+5*x^3*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/b^4/Pi^2+15/2*FresnelC(b*x) 
*FresnelS(b*x)/b^7/Pi^3+15/8*I*x^2*hypergeom([1, 1],[3/2, 2],-1/2*I*b^2*Pi 
*x^2)/b^5/Pi^3-15/8*I*x^2*hypergeom([1, 1],[3/2, 2],1/2*I*b^2*Pi*x^2)/b^5/ 
Pi^3-15*x*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/b^6/Pi^3+x^5*FresnelC(b*x)*sin 
(1/2*b^2*Pi*x^2)/b^2/Pi-7/4*x^2*sin(b^2*Pi*x^2)/b^5/Pi^3
 

3.2.82.2 Mathematica [F]

\[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx \]

input

Integrate[x^6*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x],x]
 

output

Integrate[x^6*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x], x]
 

3.2.82.3 Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.36, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {7009, 3860, 3042, 3777, 3042, 3777, 25, 3042, 3118, 7017, 3861, 3042, 3790, 15, 25, 3042, 3777, 25, 3042, 3118, 7009, 3860, 3042, 3118, 7001}

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.

\(\displaystyle \int x^6 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right ) \, dx\)

\(\Big \downarrow \) 7009

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

\(\Big \downarrow \) 3860

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 7017

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

\(\Big \downarrow \) 3861

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3790

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 7009

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

\(\Big \downarrow \) 3860

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 7001

\(\displaystyle -\frac {5 \left (\frac {3 \left (-\frac {\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )+\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b}}{\pi b^2}+\frac {x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}\right )}{\pi b^2}-\frac {x^3 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )+\frac {x^4}{4}}{2 \pi b}\right )}{\pi b^2}+\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\)

input

Int[x^6*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x],x]
 

output

(x^5*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) - (-((x^4*Cos[b^2*Pi*x^2] 
)/(b^2*Pi)) + (2*(Cos[b^2*Pi*x^2]/(b^4*Pi^2) + (x^2*Sin[b^2*Pi*x^2])/(b^2* 
Pi)))/(b^2*Pi))/(4*b*Pi) - (5*(-((x^3*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/( 
b^2*Pi)) + (3*(Cos[b^2*Pi*x^2]/(4*b^3*Pi^2) - ((FresnelC[b*x]*FresnelS[b*x 
])/(2*b) + (I/8)*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-1/2*I)*b^2*Pi 
*x^2] - (I/8)*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2]) 
/(b^2*Pi) + (x*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi)))/(b^2*Pi) + (x 
^4/4 + (Cos[b^2*Pi*x^2]/(b^4*Pi^2) + (x^2*Sin[b^2*Pi*x^2])/(b^2*Pi))/2)/(2 
*b*Pi)))/(b^2*Pi)
 

3.2.82.3.1 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 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3790

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + ((f_.)*(x_))/2]^2, x_Symbol] :> 
Simp[1/2   Int[(c + d*x)^m, x], x] - Simp[1/2   Int[(c + d*x)^m*Cos[2*e + f 
*x], x], x] /; FreeQ[{c, d, e, f, m}, x]
 

rule 3860

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ 
(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ 
(m + 1)/n], 0]))
 

rule 3861

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ 
(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ 
(m + 1)/n], 0]))
 

rule 7001

Int[FresnelC[(b_.)*(x_)]*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[b*Pi*FresnelC 
[b*x]*(FresnelS[b*x]/(4*d)), x] + (Simp[(1/8)*I*b*x^2*HypergeometricPFQ[{1, 
 1}, {3/2, 2}, (-I)*d*x^2], x] - Simp[(1/8)*I*b*x^2*HypergeometricPFQ[{1, 1 
}, {3/2, 2}, I*d*x^2], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4]
 

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 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]
 

3.2.82.4 Maple [F]

\[\int x^{6} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelC}\left (b x \right )d x\]

input

int(x^6*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x),x)
 

output

int(x^6*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x),x)
 

3.2.82.5 Fricas [F]

\[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int { x^{6} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) \,d x } \]

input

integrate(x^6*cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x),x, algorithm="fricas")
 

output

integral(x^6*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x), x)
 

3.2.82.6 Sympy [F]

\[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int x^{6} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \]

input

integrate(x**6*cos(1/2*b**2*pi*x**2)*fresnelc(b*x),x)
 

output

Integral(x**6*cos(pi*b**2*x**2/2)*fresnelc(b*x), x)
 

3.2.82.7 Maxima [F]

\[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int { x^{6} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) \,d x } \]

input

integrate(x^6*cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x),x, algorithm="maxima")
 

output

integrate(x^6*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x), x)
 

3.2.82.8 Giac [F]

\[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int { x^{6} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) \,d x } \]

input

integrate(x^6*cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x),x, algorithm="giac")
 

output

integrate(x^6*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x), x)
 

3.2.82.9 Mupad [F(-1)]

Timed out. \[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int x^6\,\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]

input

int(x^6*FresnelC(b*x)*cos((Pi*b^2*x^2)/2),x)
 

output

int(x^6*FresnelC(b*x)*cos((Pi*b^2*x^2)/2), x)