Integrand size = 20, antiderivative size = 231 \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {105 \operatorname {FresnelC}(b x)^2}{2 b^9 \pi ^4}-\frac {35 x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {40 \sin \left (b^2 \pi x^2\right )}{b^9 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3} \]
105/4*x^2/b^7/Pi^4-7/12*x^6/b^3/Pi^2-55/4*x^2*cos(b^2*Pi*x^2)/b^7/Pi^4+1/4 *x^6*cos(b^2*Pi*x^2)/b^3/Pi^2-105*x*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/b^8/ Pi^4+7*x^5*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/b^4/Pi^2+105/2*FresnelC(b*x)^ 2/b^9/Pi^4-35*x^3*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/b^6/Pi^3+x^7*FresnelC( b*x)*sin(1/2*b^2*Pi*x^2)/b^2/Pi+40*sin(b^2*Pi*x^2)/b^9/Pi^5-5/2*x^4*sin(b^ 2*Pi*x^2)/b^5/Pi^3
Time = 0.01 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00 \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {105 \operatorname {FresnelC}(b x)^2}{2 b^9 \pi ^4}-\frac {35 x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {40 \sin \left (b^2 \pi x^2\right )}{b^9 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3} \]
(105*x^2)/(4*b^7*Pi^4) - (7*x^6)/(12*b^3*Pi^2) - (55*x^2*Cos[b^2*Pi*x^2])/ (4*b^7*Pi^4) + (x^6*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (105*x*Cos[(b^2*Pi*x^2 )/2]*FresnelC[b*x])/(b^8*Pi^4) + (7*x^5*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x]) /(b^4*Pi^2) + (105*FresnelC[b*x]^2)/(2*b^9*Pi^4) - (35*x^3*FresnelC[b*x]*S in[(b^2*Pi*x^2)/2])/(b^6*Pi^3) + (x^7*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/( b^2*Pi) + (40*Sin[b^2*Pi*x^2])/(b^9*Pi^5) - (5*x^4*Sin[b^2*Pi*x^2])/(2*b^5 *Pi^3)
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.
| \(\displaystyle \int x^8 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right ) \, dx\) |
| \(\Big \downarrow \) 7009 |
| \(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^7 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
| \(\Big \downarrow \) 3860 |
| \(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^6 \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^6 \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {3 \int x^4 \cos \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {3 \int x^4 \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {3 \left (\frac {2 \int -x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}+\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\int \cos \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\int \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {7 \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 7017 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\int x^5 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3861 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\int x^4 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\int x^4 \sin \left (\frac {1}{2} b^2 \pi x^2+\frac {\pi }{2}\right )^2dx^2}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3790 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {\int x^4dx^2}{2}-\frac {1}{2} \int -x^4 \cos \left (b^2 \pi x^2\right )dx^2}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {x^6}{6}-\frac {1}{2} \int -x^4 \cos \left (b^2 \pi x^2\right )dx^2}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \int x^4 \cos \left (b^2 \pi x^2\right )dx^2+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \int x^4 \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {2 \int -x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}+\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\int \cos \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\int \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 7009 |
| \(\displaystyle -\frac {7 \left (\frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^3 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3860 |
| \(\displaystyle -\frac {7 \left (\frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \left (\frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {7 \left (\frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {\int \cos \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \left (\frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {\int \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {7 \left (\frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\) |
3.2.80.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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]
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]
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]))
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]))
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]
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]
\[\int x^{8} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelC}\left (b x \right )d x\]
Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.73 \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=-\frac {5 \, \pi ^{3} b^{6} x^{6} - 240 \, \pi b^{2} x^{2} - 3 \, {\left (\pi ^{3} b^{6} x^{6} - 55 \, \pi b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 42 \, {\left (\pi ^{3} b^{5} x^{5} - 15 \, \pi b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 315 \, \pi \operatorname {C}\left (b x\right )^{2} + 6 \, {\left (5 \, {\left (\pi ^{2} b^{4} x^{4} - 16\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{4} b^{7} x^{7} - 35 \, \pi ^{2} b^{3} x^{3}\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{6 \, \pi ^{5} b^{9}} \]
-1/6*(5*pi^3*b^6*x^6 - 240*pi*b^2*x^2 - 3*(pi^3*b^6*x^6 - 55*pi*b^2*x^2)*c os(1/2*pi*b^2*x^2)^2 - 42*(pi^3*b^5*x^5 - 15*pi*b*x)*cos(1/2*pi*b^2*x^2)*f resnel_cos(b*x) - 315*pi*fresnel_cos(b*x)^2 + 6*(5*(pi^2*b^4*x^4 - 16)*cos (1/2*pi*b^2*x^2) - (pi^4*b^7*x^7 - 35*pi^2*b^3*x^3)*fresnel_cos(b*x))*sin( 1/2*pi*b^2*x^2))/(pi^5*b^9)
Time = 14.18 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.30 \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\begin {cases} \frac {x^{7} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi b^{2}} - \frac {5 x^{6} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{6 \pi ^{2} b^{3}} - \frac {x^{6} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{3 \pi ^{2} b^{3}} + \frac {7 x^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{2} b^{4}} - \frac {5 x^{4} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{3} b^{5}} - \frac {35 x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{3} b^{6}} + \frac {40 x^{2} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{4} b^{7}} + \frac {25 x^{2} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{4} b^{7}} - \frac {105 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{4} b^{8}} + \frac {80 \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{5} b^{9}} + \frac {105 C^{2}\left (b x\right )}{2 \pi ^{4} b^{9}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**7*sin(pi*b**2*x**2/2)*fresnelc(b*x)/(pi*b**2) - 5*x**6*sin(p i*b**2*x**2/2)**2/(6*pi**2*b**3) - x**6*cos(pi*b**2*x**2/2)**2/(3*pi**2*b* *3) + 7*x**5*cos(pi*b**2*x**2/2)*fresnelc(b*x)/(pi**2*b**4) - 5*x**4*sin(p i*b**2*x**2/2)*cos(pi*b**2*x**2/2)/(pi**3*b**5) - 35*x**3*sin(pi*b**2*x**2 /2)*fresnelc(b*x)/(pi**3*b**6) + 40*x**2*sin(pi*b**2*x**2/2)**2/(pi**4*b** 7) + 25*x**2*cos(pi*b**2*x**2/2)**2/(2*pi**4*b**7) - 105*x*cos(pi*b**2*x** 2/2)*fresnelc(b*x)/(pi**4*b**8) + 80*sin(pi*b**2*x**2/2)*cos(pi*b**2*x**2/ 2)/(pi**5*b**9) + 105*fresnelc(b*x)**2/(2*pi**4*b**9), Ne(b, 0)), (0, True ))
\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int { x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) \,d x } \]
\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int { x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) \,d x } \]
Timed out. \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int x^8\,\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]