3.2.42 \(\int x^5 \operatorname {FresnelC}(b x)^2 \, dx\) [142]

3.2.42.1 Optimal result

Integrand size = 10, antiderivative size = 265 \[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\frac {5 x^4}{24 b^2 \pi ^2}+\frac {11 \cos \left (b^2 \pi x^2\right )}{6 b^6 \pi ^4}-\frac {x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{3 b^3 \pi ^2}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)^2-\frac {5 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^6 \pi ^3}-\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^5 \pi ^3}-\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {7 x^2 \sin \left (b^2 \pi x^2\right )}{12 b^4 \pi ^3} \]

output

5/24*x^4/b^2/Pi^2+11/6*cos(b^2*Pi*x^2)/b^6/Pi^4-1/12*x^4*cos(b^2*Pi*x^2)/b 
^2/Pi^2-5/3*x^3*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/b^3/Pi^2+1/6*x^6*Fresnel 
C(b*x)^2-5/2*FresnelC(b*x)*FresnelS(b*x)/b^6/Pi^3-5/8*I*x^2*hypergeom([1, 
1],[3/2, 2],-1/2*I*b^2*Pi*x^2)/b^4/Pi^3+5/8*I*x^2*hypergeom([1, 1],[3/2, 2 
],1/2*I*b^2*Pi*x^2)/b^4/Pi^3+5*x*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/b^5/Pi^ 
3-1/3*x^5*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/b/Pi+7/12*x^2*sin(b^2*Pi*x^2)/ 
b^4/Pi^3
 

3.2.42.2 Mathematica [F]

\[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\int x^5 \operatorname {FresnelC}(b x)^2 \, dx \]

input

Integrate[x^5*FresnelC[b*x]^2,x]
 

output

Integrate[x^5*FresnelC[b*x]^2, x]
 

3.2.42.3 Rubi [A] (verified)

Time = 2.23 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.34, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.600, Rules used = {6985, 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.

\(\displaystyle \int x^5 \operatorname {FresnelC}(b x)^2 \, dx\)

\(\Big \downarrow \) 6985

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

\(\Big \downarrow \) 7009

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

\(\Big \downarrow \) 3860

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 7017

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

\(\Big \downarrow \) 3861

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3790

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 7009

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

\(\Big \downarrow \) 3860

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 7001

\(\displaystyle \frac {1}{6} x^6 \operatorname {FresnelC}(b x)^2-\frac {1}{3} b \left (-\frac {5 \left (\frac {3 \left (-\frac {\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )+\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b}}{\pi b^2}+\frac {x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}\right )}{\pi b^2}-\frac {x^3 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )+\frac {x^4}{4}}{2 \pi b}\right )}{\pi b^2}+\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {2 \left (\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\)

input

Int[x^5*FresnelC[b*x]^2,x]
 

output

(x^6*FresnelC[b*x]^2)/6 - (b*((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*Si 
n[b^2*Pi*x^2])/(b^2*Pi))/2)/(2*b*Pi)))/(b^2*Pi)))/3
 

3.2.42.3.1 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 6985

Int[FresnelC[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Fresnel 
C[b*x]^2/(m + 1)), x] - Simp[2*(b/(m + 1))   Int[x^(m + 1)*Cos[(Pi/2)*b^2*x 
^2]*FresnelC[b*x], x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]
 

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]
 

3.2.42.4 Maple [F]

\[\int x^{5} \operatorname {FresnelC}\left (b x \right )^{2}d x\]

input

int(x^5*FresnelC(b*x)^2,x)
 

output

int(x^5*FresnelC(b*x)^2,x)
 

3.2.42.5 Fricas [F]

\[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\int { x^{5} \operatorname {C}\left (b x\right )^{2} \,d x } \]

input

integrate(x^5*fresnel_cos(b*x)^2,x, algorithm="fricas")
 

output

integral(x^5*fresnel_cos(b*x)^2, x)
 

3.2.42.6 Sympy [F]

\[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\int x^{5} C^{2}\left (b x\right )\, dx \]

input

integrate(x**5*fresnelc(b*x)**2,x)
 

output

Integral(x**5*fresnelc(b*x)**2, x)
 

3.2.42.7 Maxima [F]

\[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\int { x^{5} \operatorname {C}\left (b x\right )^{2} \,d x } \]

input

integrate(x^5*fresnel_cos(b*x)^2,x, algorithm="maxima")
 

output

integrate(x^5*fresnel_cos(b*x)^2, x)
 

3.2.42.8 Giac [F]

\[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\int { x^{5} \operatorname {C}\left (b x\right )^{2} \,d x } \]

input

integrate(x^5*fresnel_cos(b*x)^2,x, algorithm="giac")
 

output

integrate(x^5*fresnel_cos(b*x)^2, x)
 

3.2.42.9 Mupad [F(-1)]

Timed out. \[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\int x^5\,{\mathrm {FresnelC}\left (b\,x\right )}^2 \,d x \]

input

int(x^5*FresnelC(b*x)^2,x)
 

output

int(x^5*FresnelC(b*x)^2, x)