Integrand size = 20, antiderivative size = 185 \[ \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=-\frac {15 x^2}{4 b^5 \pi ^3}+\frac {x^6}{12 b \pi }+\frac {7 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac {15 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^6 \pi ^3}-\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }-\frac {15 \operatorname {FresnelC}(b x)^2}{2 b^7 \pi ^3}+\frac {5 x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {11 \sin \left (b^2 \pi x^2\right )}{2 b^7 \pi ^4}+\frac {x^4 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \] Output:
-15/4*x^2/b^5/Pi^3+1/12*x^6/b/Pi+7/4*x^2*cos(b^2*Pi*x^2)/b^5/Pi^3+15*x*cos (1/2*b^2*Pi*x^2)*FresnelC(b*x)/b^6/Pi^3-x^5*cos(1/2*b^2*Pi*x^2)*FresnelC(b *x)/b^2/Pi-15/2*FresnelC(b*x)^2/b^7/Pi^3+5*x^3*FresnelC(b*x)*sin(1/2*b^2*P i*x^2)/b^4/Pi^2-11/2*sin(b^2*Pi*x^2)/b^7/Pi^4+1/4*x^4*sin(b^2*Pi*x^2)/b^3/ Pi^2
Time = 0.01 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00 \[ \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=-\frac {15 x^2}{4 b^5 \pi ^3}+\frac {x^6}{12 b \pi }+\frac {7 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac {15 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^6 \pi ^3}-\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }-\frac {15 \operatorname {FresnelC}(b x)^2}{2 b^7 \pi ^3}+\frac {5 x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {11 \sin \left (b^2 \pi x^2\right )}{2 b^7 \pi ^4}+\frac {x^4 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \] Input:
Integrate[x^6*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2],x]
Output:
(-15*x^2)/(4*b^5*Pi^3) + x^6/(12*b*Pi) + (7*x^2*Cos[b^2*Pi*x^2])/(4*b^5*Pi ^3) + (15*x*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^6*Pi^3) - (x^5*Cos[(b^2* Pi*x^2)/2]*FresnelC[b*x])/(b^2*Pi) - (15*FresnelC[b*x]^2)/(2*b^7*Pi^3) + ( 5*x^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^4*Pi^2) - (11*Sin[b^2*Pi*x^2]) /(2*b^7*Pi^4) + (x^4*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)
Time = 1.92 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.54, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {7017, 3861, 3042, 3790, 15, 25, 3042, 3777, 25, 3042, 3777, 3042, 3117, 7009, 3860, 3042, 3777, 3042, 3117, 7017, 3861, 3042, 3114, 6995, 15}
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.
\(\displaystyle \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right ) \, dx\) |
\(\Big \downarrow \) 7017 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\int x^5 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3861 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\int x^4 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\int x^4 \sin \left (\frac {1}{2} b^2 \pi x^2+\frac {\pi }{2}\right )^2dx^2}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3790 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {\int x^4dx^2}{2}-\frac {1}{2} \int -x^4 \cos \left (b^2 \pi x^2\right )dx^2}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {x^6}{6}-\frac {1}{2} \int -x^4 \cos \left (b^2 \pi x^2\right )dx^2}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \int x^4 \cos \left (b^2 \pi x^2\right )dx^2+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \int x^4 \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {2 \int -x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}+\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\int \cos \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\int \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\) |
\(\Big \downarrow \) 7009 |
\(\displaystyle \frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^3 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle \frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {\int \cos \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {\int \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \frac {5 \left (-\frac {3 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\) |
\(\Big \downarrow \) 7017 |
\(\displaystyle \frac {5 \left (-\frac {3 \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\int x \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}-\frac {x \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\) |
\(\Big \downarrow \) 3861 |
\(\displaystyle \frac {5 \left (-\frac {3 \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\int \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{2 \pi b}-\frac {x \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5 \left (-\frac {3 \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\int \sin \left (\frac {1}{2} b^2 \pi x^2+\frac {\pi }{2}\right )^2dx^2}{2 \pi b}-\frac {x \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\) |
\(\Big \downarrow \) 3114 |
\(\displaystyle \frac {5 \left (-\frac {3 \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}-\frac {x \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {\sin \left (\pi b^2 x^2\right )}{2 \pi b^2}+\frac {x^2}{2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\) |
\(\Big \downarrow \) 6995 |
\(\displaystyle \frac {5 \left (-\frac {3 \left (\frac {\int \operatorname {FresnelC}(b x)d\operatorname {FresnelC}(b x)}{\pi b^3}-\frac {x \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {\sin \left (\pi b^2 x^2\right )}{2 \pi b^2}+\frac {x^2}{2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}-\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}+\frac {5 \left (\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {3 \left (\frac {\operatorname {FresnelC}(b x)^2}{2 \pi b^3}-\frac {x \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {\sin \left (\pi b^2 x^2\right )}{2 \pi b^2}+\frac {x^2}{2}}{2 \pi b}\right )}{\pi b^2}\right )}{\pi b^2}\) |
Input:
Int[x^6*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2],x]
Output:
-((x^5*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^2*Pi)) + (x^6/6 + ((x^4*Sin[b ^2*Pi*x^2])/(b^2*Pi) - (2*(-((x^2*Cos[b^2*Pi*x^2])/(b^2*Pi)) + Sin[b^2*Pi* x^2]/(b^4*Pi^2)))/(b^2*Pi))/2)/(2*b*Pi) + (5*((x^3*FresnelC[b*x]*Sin[(b^2* Pi*x^2)/2])/(b^2*Pi) - (-((x^2*Cos[b^2*Pi*x^2])/(b^2*Pi)) + Sin[b^2*Pi*x^2 ]/(b^4*Pi^2))/(4*b*Pi) - (3*(-((x*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^2* Pi)) + FresnelC[b*x]^2/(2*b^3*Pi) + (x^2/2 + Sin[b^2*Pi*x^2]/(2*b^2*Pi))/( 2*b*Pi)))/(b^2*Pi)))/(b^2*Pi)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[sin[(c_.) + ((d_.)*(x_))/2]^2, x_Symbol] :> Simp[x/2, x] - Simp[Sin[2*c + d*x]/(2*d), x] /; FreeQ[{c, d}, x]
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_)]^(n_.), x_Symbol] :> Simp[Pi*(b/( 2*d)) Subst[Int[x^n, x], x, FresnelC[b*x]], x] /; FreeQ[{b, d, n}, x] && EqQ[d^2, (Pi^2/4)*b^4]
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]
\[\int x^{6} \operatorname {FresnelC}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )d x\]
Input:
int(x^6*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2),x)
Output:
int(x^6*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2),x)
Time = 0.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.76 \[ \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {\pi ^{3} b^{6} x^{6} + 42 \, \pi b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 66 \, \pi b^{2} x^{2} - 12 \, {\left (\pi ^{3} b^{5} x^{5} - 15 \, \pi b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 90 \, \pi \operatorname {C}\left (b x\right )^{2} + 6 \, {\left (10 \, \pi ^{2} b^{3} x^{3} \operatorname {C}\left (b x\right ) + {\left (\pi ^{2} b^{4} x^{4} - 22\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{12 \, \pi ^{4} b^{7}} \] Input:
integrate(x^6*fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="fricas")
Output:
1/12*(pi^3*b^6*x^6 + 42*pi*b^2*x^2*cos(1/2*pi*b^2*x^2)^2 - 66*pi*b^2*x^2 - 12*(pi^3*b^5*x^5 - 15*pi*b*x)*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x) - 90*p i*fresnel_cos(b*x)^2 + 6*(10*pi^2*b^3*x^3*fresnel_cos(b*x) + (pi^2*b^4*x^4 - 22)*cos(1/2*pi*b^2*x^2))*sin(1/2*pi*b^2*x^2))/(pi^4*b^7)
Time = 4.32 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.43 \[ \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\begin {cases} \frac {x^{6} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{12 \pi b} + \frac {x^{6} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{12 \pi b} - \frac {x^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi b^{2}} + \frac {x^{4} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{2} b^{3}} + \frac {5 x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{2} b^{4}} - \frac {11 x^{2} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{3} b^{5}} - \frac {2 x^{2} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{3} b^{5}} + \frac {15 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{3} b^{6}} - \frac {11 \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{4} b^{7}} - \frac {15 C^{2}\left (b x\right )}{2 \pi ^{3} b^{7}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**6*fresnelc(b*x)*sin(1/2*b**2*pi*x**2),x)
Output:
Piecewise((x**6*sin(pi*b**2*x**2/2)**2/(12*pi*b) + x**6*cos(pi*b**2*x**2/2 )**2/(12*pi*b) - x**5*cos(pi*b**2*x**2/2)*fresnelc(b*x)/(pi*b**2) + x**4*s in(pi*b**2*x**2/2)*cos(pi*b**2*x**2/2)/(2*pi**2*b**3) + 5*x**3*sin(pi*b**2 *x**2/2)*fresnelc(b*x)/(pi**2*b**4) - 11*x**2*sin(pi*b**2*x**2/2)**2/(2*pi **3*b**5) - 2*x**2*cos(pi*b**2*x**2/2)**2/(pi**3*b**5) + 15*x*cos(pi*b**2* x**2/2)*fresnelc(b*x)/(pi**3*b**6) - 11*sin(pi*b**2*x**2/2)*cos(pi*b**2*x* *2/2)/(pi**4*b**7) - 15*fresnelc(b*x)**2/(2*pi**3*b**7), Ne(b, 0)), (0, Tr ue))
\[ \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x^{6} \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \] Input:
integrate(x^6*fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="maxima")
Output:
integrate(x^6*fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2), x)
\[ \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x^{6} \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \] Input:
integrate(x^6*fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="giac")
Output:
integrate(x^6*fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2), x)
Timed out. \[ \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x^6\,\mathrm {FresnelC}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \] Input:
int(x^6*FresnelC(b*x)*sin((Pi*b^2*x^2)/2),x)
Output:
int(x^6*FresnelC(b*x)*sin((Pi*b^2*x^2)/2), x)
\[ \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x^{6} \mathrm {FresnelC}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )d x \] Input:
int(x^6*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2),x)
Output:
int(x^6*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2),x)