Integrand size = 10, antiderivative size = 253 \[ \int x^7 \operatorname {FresnelS}(b x)^2 \, dx=-\frac {105 x^2}{16 b^6 \pi ^4}+\frac {7 x^6}{48 b^2 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{16 b^6 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{4 b \pi }-\frac {105 \operatorname {FresnelS}(b x)^2}{8 b^8 \pi ^4}+\frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2+\frac {105 x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {7 x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {10 \sin \left (b^2 \pi x^2\right )}{b^8 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3} \]
-105/16*x^2/b^6/Pi^4+7/48*x^6/b^2/Pi^2-55/16*x^2*cos(b^2*Pi*x^2)/b^6/Pi^4+ 1/16*x^6*cos(b^2*Pi*x^2)/b^2/Pi^2-35/4*x^3*cos(1/2*b^2*Pi*x^2)*FresnelS(b* x)/b^5/Pi^3+1/4*x^7*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b/Pi-105/8*FresnelS( b*x)^2/b^8/Pi^4+1/8*x^8*FresnelS(b*x)^2+105/4*x*FresnelS(b*x)*sin(1/2*b^2* Pi*x^2)/b^7/Pi^4-7/4*x^5*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^3/Pi^2+10*sin (b^2*Pi*x^2)/b^8/Pi^5-5/8*x^4*sin(b^2*Pi*x^2)/b^4/Pi^3
Time = 0.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.72 \[ \int x^7 \operatorname {FresnelS}(b x)^2 \, dx=\frac {-315 b^2 \pi x^2+7 b^6 \pi ^3 x^6+3 b^2 \pi x^2 \left (-55+b^4 \pi ^2 x^4\right ) \cos \left (b^2 \pi x^2\right )+6 \pi \left (-105+b^8 \pi ^4 x^8\right ) \operatorname {FresnelS}(b x)^2+12 b \pi x \operatorname {FresnelS}(b x) \left (b^2 \pi x^2 \left (-35+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )-7 \left (-15+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+480 \sin \left (b^2 \pi x^2\right )-30 b^4 \pi ^2 x^4 \sin \left (b^2 \pi x^2\right )}{48 b^8 \pi ^5} \]
(-315*b^2*Pi*x^2 + 7*b^6*Pi^3*x^6 + 3*b^2*Pi*x^2*(-55 + b^4*Pi^2*x^4)*Cos[ b^2*Pi*x^2] + 6*Pi*(-105 + b^8*Pi^4*x^8)*FresnelS[b*x]^2 + 12*b*Pi*x*Fresn elS[b*x]*(b^2*Pi*x^2*(-35 + b^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2] - 7*(-15 + b ^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2]) + 480*Sin[b^2*Pi*x^2] - 30*b^4*Pi^2*x^4* Sin[b^2*Pi*x^2])/(48*b^8*Pi^5)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^7 \operatorname {FresnelS}(b x)^2 \, dx\) |
| \(\Big \downarrow \) 6984 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \int x^8 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx\) |
| \(\Big \downarrow \) 7008 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^7 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3860 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^6 \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^6 \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {3 \int x^4 \cos \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {3 \int x^4 \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {3 \left (\frac {2 \int -x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}+\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\int \cos \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\int \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 7016 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \int x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^5 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3860 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \int x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^4 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{2 \pi b}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \int x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right )^2dx^2}{2 \pi b}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3790 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \int x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {\int x^4dx^2}{2}-\frac {1}{2} \int x^4 \cos \left (b^2 \pi x^2\right )dx^2}{2 \pi b}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \int x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^6}{6}-\frac {1}{2} \int x^4 \cos \left (b^2 \pi x^2\right )dx^2}{2 \pi b}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \int x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^6}{6}-\frac {1}{2} \int x^4 \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{2 \pi b}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \int x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (-\frac {2 \int -x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \int x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {2 \int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \int x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {2 \int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \int x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {2 \left (\frac {\int \cos \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \int x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {2 \left (\frac {\int \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \int x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 7008 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \left (\frac {3 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^3 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}-\frac {x^3 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3860 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \left (\frac {3 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}-\frac {x^3 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \left (\frac {3 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^2 \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}-\frac {x^3 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \left (\frac {3 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {\int \cos \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^3 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \left (\frac {3 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {\int \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}-\frac {x^3 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)^2-\frac {1}{4} b \left (\frac {7 \left (-\frac {5 \left (\frac {3 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}-\frac {x^3 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )}{\pi b^2}+\frac {x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^6}{6}}{2 \pi b}\right )}{\pi b^2}-\frac {x^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x^4 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{\pi b^2}}{4 \pi b}\right )\) |
3.1.31.3.1 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[FresnelS[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Fresnel S[b*x]^2/(m + 1)), x] - Simp[2*(b/(m + 1)) Int[x^(m + 1)*Sin[(Pi/2)*b^2*x ^2]*FresnelS[b*x], x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]
Int[FresnelS[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[(-x ^(m - 1))*Cos[d*x^2]*(FresnelS[b*x]/(2*d)), x] + (Simp[(m - 1)/(2*d) Int[ x^(m - 2)*Cos[d*x^2]*FresnelS[b*x], x], x] + Simp[1/(2*b*Pi) Int[x^(m - 1 )*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IG tQ[m, 1]
Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^( m - 1)*Sin[d*x^2]*(FresnelS[b*x]/(2*d)), x] + (-Simp[1/(Pi*b) Int[x^(m - 1)*Sin[d*x^2]^2, x], x] - Simp[(m - 1)/(2*d) Int[x^(m - 2)*Sin[d*x^2]*Fre snelS[b*x], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IGtQ[m , 1]
\[\int x^{7} \operatorname {FresnelS}\left (b x \right )^{2}d x\]
Time = 0.26 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.72 \[ \int x^7 \operatorname {FresnelS}(b x)^2 \, dx=\frac {2 \, \pi ^{3} b^{6} x^{6} - 75 \, \pi b^{2} x^{2} + 3 \, {\left (\pi ^{3} b^{6} x^{6} - 55 \, \pi b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 6 \, {\left (\pi ^{4} b^{7} x^{7} - 35 \, \pi ^{2} b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - 3 \, {\left (105 \, \pi - \pi ^{5} b^{8} x^{8}\right )} \operatorname {S}\left (b x\right )^{2} - 6 \, {\left (5 \, {\left (\pi ^{2} b^{4} x^{4} - 16\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 7 \, {\left (\pi ^{3} b^{5} x^{5} - 15 \, \pi b x\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{24 \, \pi ^{5} b^{8}} \]
1/24*(2*pi^3*b^6*x^6 - 75*pi*b^2*x^2 + 3*(pi^3*b^6*x^6 - 55*pi*b^2*x^2)*co s(1/2*pi*b^2*x^2)^2 + 6*(pi^4*b^7*x^7 - 35*pi^2*b^3*x^3)*cos(1/2*pi*b^2*x^ 2)*fresnel_sin(b*x) - 3*(105*pi - pi^5*b^8*x^8)*fresnel_sin(b*x)^2 - 6*(5* (pi^2*b^4*x^4 - 16)*cos(1/2*pi*b^2*x^2) + 7*(pi^3*b^5*x^5 - 15*pi*b*x)*fre snel_sin(b*x))*sin(1/2*pi*b^2*x^2))/(pi^5*b^8)
\[ \int x^7 \operatorname {FresnelS}(b x)^2 \, dx=\int x^{7} S^{2}\left (b x\right )\, dx \]
\[ \int x^7 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{7} \operatorname {S}\left (b x\right )^{2} \,d x } \]
\[ \int x^7 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{7} \operatorname {S}\left (b x\right )^{2} \,d x } \]
Timed out. \[ \int x^7 \operatorname {FresnelS}(b x)^2 \, dx=\int x^7\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \]