Integrand size = 8, antiderivative size = 124 \[ \int x^7 \operatorname {FresnelS}(b x) \, dx=-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }-\frac {105 \operatorname {FresnelS}(b x)}{8 b^8 \pi ^4}+\frac {1}{8} x^8 \operatorname {FresnelS}(b x)+\frac {105 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {7 x^5 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2} \]
-35/8*x^3*cos(1/2*b^2*Pi*x^2)/b^5/Pi^3+1/8*x^7*cos(1/2*b^2*Pi*x^2)/b/Pi-10 5/8*FresnelS(b*x)/b^8/Pi^4+1/8*x^8*FresnelS(b*x)+105/8*x*sin(1/2*b^2*Pi*x^ 2)/b^7/Pi^4-7/8*x^5*sin(1/2*b^2*Pi*x^2)/b^3/Pi^2
Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.71 \[ \int x^7 \operatorname {FresnelS}(b x) \, dx=\frac {b^3 \pi x^3 \left (-35+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )+\left (-105+b^8 \pi ^4 x^8\right ) \operatorname {FresnelS}(b x)-7 b x \left (-15+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^8 \pi ^4} \]
(b^3*Pi*x^3*(-35 + b^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2] + (-105 + b^8*Pi^4*x^ 8)*FresnelS[b*x] - 7*b*x*(-15 + b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2])/(8*b^8* Pi^4)
Time = 0.51 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6980, 3866, 3867, 3866, 3867, 3832}
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.
\(\displaystyle \int x^7 \operatorname {FresnelS}(b x) \, dx\) |
\(\Big \downarrow \) 6980 |
\(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)-\frac {1}{8} b \int x^8 \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx\) |
\(\Big \downarrow \) 3866 |
\(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)-\frac {1}{8} b \left (\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {x^7 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3867 |
\(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)-\frac {1}{8} b \left (\frac {7 \left (\frac {x^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {5 \int x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3866 |
\(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)-\frac {1}{8} b \left (\frac {7 \left (\frac {x^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {5 \left (\frac {3 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {x^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3867 |
\(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)-\frac {1}{8} b \left (\frac {7 \left (\frac {x^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\int \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}\right )}{\pi b^2}-\frac {x^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelS}(b x)-\frac {1}{8} b \left (\frac {7 \left (\frac {x^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\operatorname {FresnelS}(b x)}{\pi b^3}\right )}{\pi b^2}-\frac {x^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^7 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
(x^8*FresnelS[b*x])/8 - (b*(-((x^7*Cos[(b^2*Pi*x^2)/2])/(b^2*Pi)) + (7*((x ^5*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) - (5*(-((x^3*Cos[(b^2*Pi*x^2)/2])/(b^2*Pi )) + (3*(-(FresnelS[b*x]/(b^3*Pi)) + (x*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi)))/(b ^2*Pi)))/(b^2*Pi)))/(b^2*Pi)))/8
3.1.1.3.1 Defintions of rubi rules used
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(-e^ (n - 1))*(e*x)^(m - n + 1)*(Cos[c + d*x^n]/(d*n)), x] + Simp[e^n*((m - n + 1)/(d*n)) Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]
Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(Sin[c + d*x^n]/(d*n)), x] - Simp[e^n*((m - n + 1)/ (d*n)) Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]
Int[FresnelS[(b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1 )*(FresnelS[b*x]/(d*(m + 1))), x] - Simp[b/(d*(m + 1)) Int[(d*x)^(m + 1)* Sin[(Pi/2)*b^2*x^2], x], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.23
method | result | size |
meijerg | \(\frac {\pi \,b^{3} x^{11} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {11}{4}\right ], \left [\frac {3}{2}, \frac {7}{4}, \frac {15}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{66}\) | \(29\) |
derivativedivides | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right ) b^{8} x^{8}}{8}+\frac {b^{7} x^{7} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }-\frac {7 \left (\frac {b^{5} x^{5} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {5 \left (-\frac {b^{3} x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\frac {3 b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {3 \,\operatorname {FresnelS}\left (b x \right )}{\pi }}{\pi }\right )}{\pi }\right )}{8 \pi }}{b^{8}}\) | \(123\) |
default | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right ) b^{8} x^{8}}{8}+\frac {b^{7} x^{7} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }-\frac {7 \left (\frac {b^{5} x^{5} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {5 \left (-\frac {b^{3} x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\frac {3 b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {3 \,\operatorname {FresnelS}\left (b x \right )}{\pi }}{\pi }\right )}{\pi }\right )}{8 \pi }}{b^{8}}\) | \(123\) |
parts | \(\frac {x^{8} \operatorname {FresnelS}\left (b x \right )}{8}-\frac {b \left (-\frac {x^{7} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {\frac {7 x^{5} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {35 \left (-\frac {x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {\frac {3 x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {3 \,\operatorname {FresnelS}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}}{b^{2} \pi }\right )}{b^{2} \pi }}{b^{2} \pi }\right )}{8}\) | \(153\) |
Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.68 \[ \int x^7 \operatorname {FresnelS}(b x) \, dx=\frac {{\left (\pi ^{3} b^{7} x^{7} - 35 \, \pi b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{4} b^{8} x^{8} - 105\right )} \operatorname {S}\left (b x\right ) - 7 \, {\left (\pi ^{2} b^{5} x^{5} - 15 \, b x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{8 \, \pi ^{4} b^{8}} \]
1/8*((pi^3*b^7*x^7 - 35*pi*b^3*x^3)*cos(1/2*pi*b^2*x^2) + (pi^4*b^8*x^8 - 105)*fresnel_sin(b*x) - 7*(pi^2*b^5*x^5 - 15*b*x)*sin(1/2*pi*b^2*x^2))/(pi ^4*b^8)
Time = 1.16 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.48 \[ \int x^7 \operatorname {FresnelS}(b x) \, dx=\frac {231 x^{8} S\left (b x\right ) \Gamma \left (\frac {3}{4}\right )}{512 \Gamma \left (\frac {15}{4}\right )} + \frac {231 x^{7} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{512 \pi b \Gamma \left (\frac {15}{4}\right )} - \frac {1617 x^{5} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{512 \pi ^{2} b^{3} \Gamma \left (\frac {15}{4}\right )} - \frac {8085 x^{3} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{512 \pi ^{3} b^{5} \Gamma \left (\frac {15}{4}\right )} + \frac {24255 x \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{512 \pi ^{4} b^{7} \Gamma \left (\frac {15}{4}\right )} - \frac {24255 S\left (b x\right ) \Gamma \left (\frac {3}{4}\right )}{512 \pi ^{4} b^{8} \Gamma \left (\frac {15}{4}\right )} \]
231*x**8*fresnels(b*x)*gamma(3/4)/(512*gamma(15/4)) + 231*x**7*cos(pi*b**2 *x**2/2)*gamma(3/4)/(512*pi*b*gamma(15/4)) - 1617*x**5*sin(pi*b**2*x**2/2) *gamma(3/4)/(512*pi**2*b**3*gamma(15/4)) - 8085*x**3*cos(pi*b**2*x**2/2)*g amma(3/4)/(512*pi**3*b**5*gamma(15/4)) + 24255*x*sin(pi*b**2*x**2/2)*gamma (3/4)/(512*pi**4*b**7*gamma(15/4)) - 24255*fresnels(b*x)*gamma(3/4)/(512*p i**4*b**8*gamma(15/4))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02 \[ \int x^7 \operatorname {FresnelS}(b x) \, dx=\frac {1}{8} \, x^{8} \operatorname {S}\left (b x\right ) - \frac {\sqrt {\frac {1}{2}} {\left (\left (105 i + 105\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) - \left (105 i - 105\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right ) - 4 \, {\left (\sqrt {\frac {1}{2}} \pi ^{4} b^{7} x^{7} - 35 \, \sqrt {\frac {1}{2}} \pi ^{2} b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 28 \, {\left (\sqrt {\frac {1}{2}} \pi ^{3} b^{5} x^{5} - 15 \, \sqrt {\frac {1}{2}} \pi b x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )}}{16 \, \pi ^{5} b^{8}} \]
1/8*x^8*fresnel_sin(b*x) - 1/16*sqrt(1/2)*((105*I + 105)*(1/4)^(1/4)*pi*er f(sqrt(1/2*I*pi)*b*x) - (105*I - 105)*(1/4)^(1/4)*pi*erf(sqrt(-1/2*I*pi)*b *x) - 4*(sqrt(1/2)*pi^4*b^7*x^7 - 35*sqrt(1/2)*pi^2*b^3*x^3)*cos(1/2*pi*b^ 2*x^2) + 28*(sqrt(1/2)*pi^3*b^5*x^5 - 15*sqrt(1/2)*pi*b*x)*sin(1/2*pi*b^2* x^2))/(pi^5*b^8)
\[ \int x^7 \operatorname {FresnelS}(b x) \, dx=\int { x^{7} \operatorname {S}\left (b x\right ) \,d x } \]
Timed out. \[ \int x^7 \operatorname {FresnelS}(b x) \, dx=\int x^7\,\mathrm {FresnelS}\left (b\,x\right ) \,d x \]