Integrand size = 8, antiderivative size = 109 \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}+\frac {1}{7} x^7 \operatorname {FresnelC}(b x)+\frac {24 x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}-\frac {x^6 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b \pi } \]
48/7*cos(1/2*b^2*Pi*x^2)/b^7/Pi^4-6/7*x^4*cos(1/2*b^2*Pi*x^2)/b^3/Pi^2+1/7 *x^7*FresnelC(b*x)+24/7*x^2*sin(1/2*b^2*Pi*x^2)/b^5/Pi^3-1/7*x^6*sin(1/2*b ^2*Pi*x^2)/b/Pi
Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.76 \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=-\frac {6 \left (-8+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}+\frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {x^2 \left (-24+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3} \]
(-6*(-8 + b^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2])/(7*b^7*Pi^4) + (x^7*FresnelC[ b*x])/7 - (x^2*(-24 + b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2])/(7*b^5*Pi^3)
Time = 0.57 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {6981, 3861, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3118}
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.
| \(\displaystyle \int x^6 \operatorname {FresnelC}(b x) \, dx\) |
| \(\Big \downarrow \) 6981 |
| \(\displaystyle \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{7} b \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx\) |
| \(\Big \downarrow \) 3861 |
| \(\displaystyle \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{14} b \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx^2\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{14} b \int x^6 \sin \left (\frac {1}{2} b^2 \pi x^2+\frac {\pi }{2}\right )dx^2\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{14} b \left (\frac {6 \int -x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{\pi b^2}+\frac {2 x^6 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{14} b \left (\frac {2 x^6 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {6 \int x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{14} b \left (\frac {2 x^6 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {6 \int x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{14} b \left (\frac {2 x^6 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {6 \left (\frac {4 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {2 x^4 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{14} b \left (\frac {2 x^6 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {6 \left (\frac {4 \int x^2 \sin \left (\frac {1}{2} b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{\pi b^2}-\frac {2 x^4 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{14} b \left (\frac {2 x^6 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {6 \left (\frac {4 \left (\frac {2 \int -\sin \left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{\pi b^2}+\frac {2 x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {2 x^4 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{14} b \left (\frac {2 x^6 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {6 \left (\frac {4 \left (\frac {2 x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {2 \int \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{\pi b^2}\right )}{\pi b^2}-\frac {2 x^4 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{14} b \left (\frac {2 x^6 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {6 \left (\frac {4 \left (\frac {2 x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {2 \int \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{\pi b^2}\right )}{\pi b^2}-\frac {2 x^4 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{14} b \left (\frac {2 x^6 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {6 \left (\frac {4 \left (\frac {2 x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {4 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}\right )}{\pi b^2}-\frac {2 x^4 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\) |
(x^7*FresnelC[b*x])/7 - (b*((2*x^6*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) - (6*((-2 *x^4*Cos[(b^2*Pi*x^2)/2])/(b^2*Pi) + (4*((4*Cos[(b^2*Pi*x^2)/2])/(b^4*Pi^2 ) + (2*x^2*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi)))/(b^2*Pi)))/(b^2*Pi)))/14
3.2.11.3.1 Defintions of rubi rules used
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[((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[FresnelC[(b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1 )*(FresnelC[b*x]/(d*(m + 1))), x] - Simp[b/(d*(m + 1)) Int[(d*x)^(m + 1)* Cos[(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 = 0.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.24
| method | result | size |
| meijerg | \(\frac {b \,x^{8} \operatorname {hypergeom}\left (\left [\frac {1}{4}, 2\right ], \left [\frac {1}{2}, \frac {5}{4}, 3\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{8}\) | \(26\) |
| derivativedivides | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{7} x^{7}}{7}-\frac {b^{6} x^{6} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 \pi }+\frac {-\frac {6 b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 \pi }+\frac {6 \left (\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{7 \pi }}{\pi }}{b^{7}}\) | \(107\) |
| default | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{7} x^{7}}{7}-\frac {b^{6} x^{6} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 \pi }+\frac {-\frac {6 b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 \pi }+\frac {6 \left (\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{7 \pi }}{\pi }}{b^{7}}\) | \(107\) |
| parts | \(\frac {x^{7} \operatorname {FresnelC}\left (b x \right )}{7}-\frac {b \left (\frac {x^{6} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {6 \left (-\frac {x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {\frac {4 x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{4} \pi ^{2}}}{b^{2} \pi }\right )}{b^{2} \pi }\right )}{7}\) | \(112\) |
Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.72 \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=\frac {\pi ^{4} b^{7} x^{7} \operatorname {C}\left (b x\right ) - 6 \, {\left (\pi ^{2} b^{4} x^{4} - 8\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{3} b^{6} x^{6} - 24 \, \pi b^{2} x^{2}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{7 \, \pi ^{4} b^{7}} \]
1/7*(pi^4*b^7*x^7*fresnel_cos(b*x) - 6*(pi^2*b^4*x^4 - 8)*cos(1/2*pi*b^2*x ^2) - (pi^3*b^6*x^6 - 24*pi*b^2*x^2)*sin(1/2*pi*b^2*x^2))/(pi^4*b^7)
Time = 1.17 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.40 \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=\frac {x^{7} C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{28 \Gamma \left (\frac {5}{4}\right )} - \frac {x^{6} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{28 \pi b \Gamma \left (\frac {5}{4}\right )} - \frac {3 x^{4} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{14 \pi ^{2} b^{3} \Gamma \left (\frac {5}{4}\right )} + \frac {6 x^{2} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{7 \pi ^{3} b^{5} \Gamma \left (\frac {5}{4}\right )} + \frac {12 \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{7 \pi ^{4} b^{7} \Gamma \left (\frac {5}{4}\right )} \]
x**7*fresnelc(b*x)*gamma(1/4)/(28*gamma(5/4)) - x**6*sin(pi*b**2*x**2/2)*g amma(1/4)/(28*pi*b*gamma(5/4)) - 3*x**4*cos(pi*b**2*x**2/2)*gamma(1/4)/(14 *pi**2*b**3*gamma(5/4)) + 6*x**2*sin(pi*b**2*x**2/2)*gamma(1/4)/(7*pi**3*b **5*gamma(5/4)) + 12*cos(pi*b**2*x**2/2)*gamma(1/4)/(7*pi**4*b**7*gamma(5/ 4))
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.68 \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=\frac {1}{7} \, x^{7} \operatorname {C}\left (b x\right ) - \frac {6 \, {\left (\pi ^{2} b^{4} x^{4} - 8\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{3} b^{6} x^{6} - 24 \, \pi b^{2} x^{2}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{7 \, \pi ^{4} b^{7}} \]
1/7*x^7*fresnel_cos(b*x) - 1/7*(6*(pi^2*b^4*x^4 - 8)*cos(1/2*pi*b^2*x^2) + (pi^3*b^6*x^6 - 24*pi*b^2*x^2)*sin(1/2*pi*b^2*x^2))/(pi^4*b^7)
\[ \int x^6 \operatorname {FresnelC}(b x) \, dx=\int { x^{6} \operatorname {C}\left (b x\right ) \,d x } \]
Timed out. \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=\int x^6\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]