Integrand size = 8, antiderivative size = 99 \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=-\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b^3 \pi ^2}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)-\frac {5 \operatorname {FresnelS}(b x)}{2 b^6 \pi ^3}+\frac {5 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac {x^5 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b \pi } \]
-5/6*x^3*cos(1/2*b^2*Pi*x^2)/b^3/Pi^2+1/6*x^6*FresnelC(b*x)-5/2*FresnelS(b *x)/b^6/Pi^3+5/2*x*sin(1/2*b^2*Pi*x^2)/b^5/Pi^3-1/6*x^5*sin(1/2*b^2*Pi*x^2 )/b/Pi
Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.81 \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=\frac {-5 b^3 \pi x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )+b^6 \pi ^3 x^6 \operatorname {FresnelC}(b x)-15 \operatorname {FresnelS}(b x)+b x \left (15-b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b^6 \pi ^3} \]
(-5*b^3*Pi*x^3*Cos[(b^2*Pi*x^2)/2] + b^6*Pi^3*x^6*FresnelC[b*x] - 15*Fresn elS[b*x] + b*x*(15 - b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2])/(6*b^6*Pi^3)
Time = 0.40 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6981, 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.
\(\displaystyle \int x^5 \operatorname {FresnelC}(b x) \, dx\) |
\(\Big \downarrow \) 6981 |
\(\displaystyle \frac {1}{6} x^6 \operatorname {FresnelC}(b x)-\frac {1}{6} b \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx\) |
\(\Big \downarrow \) 3867 |
\(\displaystyle \frac {1}{6} x^6 \operatorname {FresnelC}(b x)-\frac {1}{6} b \left (\frac {x^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {5 \int x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3866 |
\(\displaystyle \frac {1}{6} x^6 \operatorname {FresnelC}(b x)-\frac {1}{6} b \left (\frac {x^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {5 \left (\frac {3 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {x^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3867 |
\(\displaystyle \frac {1}{6} x^6 \operatorname {FresnelC}(b x)-\frac {1}{6} b \left (\frac {x^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\int \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}\right )}{\pi b^2}-\frac {x^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {1}{6} x^6 \operatorname {FresnelC}(b x)-\frac {1}{6} b \left (\frac {x^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {5 \left (\frac {3 \left (\frac {x \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\operatorname {FresnelS}(b x)}{\pi b^3}\right )}{\pi b^2}-\frac {x^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\) |
(x^6*FresnelC[b*x])/6 - (b*((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)))/6
3.2.12.3.1 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[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 = 0.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.26
method | result | size |
meijerg | \(\frac {b \,x^{7} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {1}{2}, \frac {5}{4}, \frac {11}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{7}\) | \(26\) |
derivativedivides | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{6} x^{6}}{6}-\frac {b^{5} x^{5} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 \pi }+\frac {-\frac {5 b^{3} x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 \pi }+\frac {5 \left (\frac {3 b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {3 \,\operatorname {FresnelS}\left (b x \right )}{\pi }\right )}{6 \pi }}{\pi }}{b^{6}}\) | \(97\) |
default | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{6} x^{6}}{6}-\frac {b^{5} x^{5} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 \pi }+\frac {-\frac {5 b^{3} x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 \pi }+\frac {5 \left (\frac {3 b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {3 \,\operatorname {FresnelS}\left (b x \right )}{\pi }\right )}{6 \pi }}{\pi }}{b^{6}}\) | \(97\) |
parts | \(\frac {x^{6} \operatorname {FresnelC}\left (b x \right )}{6}-\frac {b \left (\frac {x^{5} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {5 \left (-\frac {x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {\frac {3 x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {3 \,\operatorname {FresnelS}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}}{b^{2} \pi }\right )}{b^{2} \pi }\right )}{6}\) | \(123\) |
Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.87 \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=\frac {\pi ^{3} b^{7} x^{6} \operatorname {C}\left (b x\right ) - 5 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{6} x^{5} - 15 \, b^{2} x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 15 \, \sqrt {b^{2}} \operatorname {S}\left (\sqrt {b^{2}} x\right )}{6 \, \pi ^{3} b^{7}} \]
1/6*(pi^3*b^7*x^6*fresnel_cos(b*x) - 5*pi*b^4*x^3*cos(1/2*pi*b^2*x^2) - (p i^2*b^6*x^5 - 15*b^2*x)*sin(1/2*pi*b^2*x^2) - 15*sqrt(b^2)*fresnel_sin(sqr t(b^2)*x))/(pi^3*b^7)
Time = 0.49 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.49 \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=\frac {b x^{7} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{4}, \frac {7}{4} \\ \frac {1}{2}, \frac {5}{4}, \frac {11}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{16 \Gamma \left (\frac {5}{4}\right ) \Gamma \left (\frac {11}{4}\right )} \]
b*x**7*gamma(1/4)*gamma(7/4)*hyper((1/4, 7/4), (1/2, 5/4, 11/4), -pi**2*b* *4*x**4/16)/(16*gamma(5/4)*gamma(11/4))
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.11 \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=\frac {1}{6} \, x^{6} \operatorname {C}\left (b x\right ) - \frac {\sqrt {\frac {1}{2}} {\left (20 \, \sqrt {\frac {1}{2}} \pi ^{2} b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \left (15 i + 15\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) - \left (15 i - 15\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right ) + 4 \, {\left (\sqrt {\frac {1}{2}} \pi ^{3} b^{5} x^{5} - 15 \, \sqrt {\frac {1}{2}} \pi b x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )}}{12 \, \pi ^{4} b^{6}} \]
1/6*x^6*fresnel_cos(b*x) - 1/12*sqrt(1/2)*(20*sqrt(1/2)*pi^2*b^3*x^3*cos(1 /2*pi*b^2*x^2) + (15*I + 15)*(1/4)^(1/4)*pi*erf(sqrt(1/2*I*pi)*b*x) - (15* I - 15)*(1/4)^(1/4)*pi*erf(sqrt(-1/2*I*pi)*b*x) + 4*(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^4*b^6)
\[ \int x^5 \operatorname {FresnelC}(b x) \, dx=\int { x^{5} \operatorname {C}\left (b x\right ) \,d x } \]
Timed out. \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=\int x^5\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]