\(\int x^7 \operatorname {FresnelC}(b x) \, dx\) [110]

Optimal result

Integrand size = 8, antiderivative size = 124 \[ \int x^7 \operatorname {FresnelC}(b x) \, dx=\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}-\frac {105 \operatorname {FresnelC}(b x)}{8 b^8 \pi ^4}+\frac {1}{8} x^8 \operatorname {FresnelC}(b x)+\frac {35 x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi } \] Output:

105/8*x*cos(1/2*b^2*Pi*x^2)/b^7/Pi^4-7/8*x^5*cos(1/2*b^2*Pi*x^2)/b^3/Pi^2- 
105/8*FresnelC(b*x)/b^8/Pi^4+1/8*x^8*FresnelC(b*x)+35/8*x^3*sin(1/2*b^2*Pi 
*x^2)/b^5/Pi^3-1/8*x^7*sin(1/2*b^2*Pi*x^2)/b/Pi
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.72 \[ \int x^7 \operatorname {FresnelC}(b x) \, dx=\frac {-7 b x \left (-15+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )+\left (-105+b^8 \pi ^4 x^8\right ) \operatorname {FresnelC}(b x)+b^3 \pi x^3 \left (35-b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^8 \pi ^4} \] Input:

Integrate[x^7*FresnelC[b*x],x]
 

Output:

(-7*b*x*(-15 + b^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2] + (-105 + b^8*Pi^4*x^8)*F 
resnelC[b*x] + b^3*Pi*x^3*(35 - b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2])/(8*b^8* 
Pi^4)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6981, 3867, 3866, 3867, 3866, 3833}

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.

\(\displaystyle \int x^7 \operatorname {FresnelC}(b x) \, dx\)

\(\Big \downarrow \) 6981

\(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelC}(b x)-\frac {1}{8} b \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx\)

\(\Big \downarrow \) 3867

\(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelC}(b x)-\frac {1}{8} b \left (\frac {x^7 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {7 \int x^6 \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}\right )\)

\(\Big \downarrow \) 3866

\(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelC}(b x)-\frac {1}{8} b \left (\frac {x^7 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {7 \left (\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {x^5 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3867

\(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelC}(b x)-\frac {1}{8} b \left (\frac {x^7 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {7 \left (\frac {5 \left (\frac {x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {3 \int x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3866

\(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelC}(b x)-\frac {1}{8} b \left (\frac {x^7 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {7 \left (\frac {5 \left (\frac {x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {3 \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {1}{8} x^8 \operatorname {FresnelC}(b x)-\frac {1}{8} b \left (\frac {x^7 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {7 \left (\frac {5 \left (\frac {x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {3 \left (\frac {\operatorname {FresnelC}(b x)}{\pi b^3}-\frac {x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^5 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\)

Input:

Int[x^7*FresnelC[b*x],x]
 

Output:

(x^8*FresnelC[b*x])/8 - (b*((x^7*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) - (7*(-((x^ 
5*Cos[(b^2*Pi*x^2)/2])/(b^2*Pi)) + (5*((-3*(-((x*Cos[(b^2*Pi*x^2)/2])/(b^2 
*Pi)) + FresnelC[b*x]/(b^3*Pi)))/(b^2*Pi) + (x^3*Sin[(b^2*Pi*x^2)/2])/(b^2 
*Pi)))/(b^2*Pi)))/(b^2*Pi)))/8
 

Defintions of rubi rules used

rule 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3866

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]
 

rule 3867

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]
 

rule 6981

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 0.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.21

method	result	size
meijerg	\(\frac {b \,x^{9} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {9}{4}\right ], \left [\frac {1}{2}, \frac {5}{4}, \frac {13}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{9}\)	\(26\)
derivativedivides	\(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{8} x^{8}}{8}-\frac {b^{7} x^{7} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }+\frac {-\frac {7 b^{5} x^{5} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }+\frac {7 \left (\frac {5 b^{3} x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {15 \left (-\frac {b x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\operatorname {FresnelC}\left (b x \right )}{\pi }\right )}{\pi }\right )}{8 \pi }}{\pi }}{b^{8}}\)	\(123\)
default	\(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{8} x^{8}}{8}-\frac {b^{7} x^{7} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }+\frac {-\frac {7 b^{5} x^{5} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }+\frac {7 \left (\frac {5 b^{3} x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {15 \left (-\frac {b x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\operatorname {FresnelC}\left (b x \right )}{\pi }\right )}{\pi }\right )}{8 \pi }}{\pi }}{b^{8}}\)	\(123\)
parts	\(\frac {x^{8} \operatorname {FresnelC}\left (b x \right )}{8}-\frac {b \left (\frac {x^{7} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {7 \left (-\frac {x^{5} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {\frac {5 x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {15 \left (-\frac {x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {\operatorname {FresnelC}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b^{2} \pi }}{b^{2} \pi }\right )}{b^{2} \pi }\right )}{8}\)	\(152\)

Input:

int(x^7*FresnelC(b*x),x,method=_RETURNVERBOSE)
 

Output:

1/9*b*x^9*hypergeom([1/4,9/4],[1/2,5/4,13/4],-1/16*x^4*Pi^2*b^4)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.69 \[ \int x^7 \operatorname {FresnelC}(b x) \, dx=-\frac {7 \, {\left (\pi ^{2} b^{5} x^{5} - 15 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{4} b^{8} x^{8} - 105\right )} \operatorname {C}\left (b x\right ) + {\left (\pi ^{3} b^{7} x^{7} - 35 \, \pi b^{3} x^{3}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{8 \, \pi ^{4} b^{8}} \] Input:

integrate(x^7*fresnel_cos(b*x),x, algorithm="fricas")
 

Output:

-1/8*(7*(pi^2*b^5*x^5 - 15*b*x)*cos(1/2*pi*b^2*x^2) - (pi^4*b^8*x^8 - 105) 
*fresnel_cos(b*x) + (pi^3*b^7*x^7 - 35*pi*b^3*x^3)*sin(1/2*pi*b^2*x^2))/(p 
i^4*b^8)
 

Sympy [A] (verification not implemented)

Time = 1.19 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.48 \[ \int x^7 \operatorname {FresnelC}(b x) \, dx=\frac {45 x^{8} C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{512 \Gamma \left (\frac {13}{4}\right )} - \frac {45 x^{7} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{512 \pi b \Gamma \left (\frac {13}{4}\right )} - \frac {315 x^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{512 \pi ^{2} b^{3} \Gamma \left (\frac {13}{4}\right )} + \frac {1575 x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{512 \pi ^{3} b^{5} \Gamma \left (\frac {13}{4}\right )} + \frac {4725 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{512 \pi ^{4} b^{7} \Gamma \left (\frac {13}{4}\right )} - \frac {4725 C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{512 \pi ^{4} b^{8} \Gamma \left (\frac {13}{4}\right )} \] Input:

integrate(x**7*fresnelc(b*x),x)
 

Output:

45*x**8*fresnelc(b*x)*gamma(1/4)/(512*gamma(13/4)) - 45*x**7*sin(pi*b**2*x 
**2/2)*gamma(1/4)/(512*pi*b*gamma(13/4)) - 315*x**5*cos(pi*b**2*x**2/2)*ga 
mma(1/4)/(512*pi**2*b**3*gamma(13/4)) + 1575*x**3*sin(pi*b**2*x**2/2)*gamm 
a(1/4)/(512*pi**3*b**5*gamma(13/4)) + 4725*x*cos(pi*b**2*x**2/2)*gamma(1/4 
)/(512*pi**4*b**7*gamma(13/4)) - 4725*fresnelc(b*x)*gamma(1/4)/(512*pi**4* 
b**8*gamma(13/4))
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02 \[ \int x^7 \operatorname {FresnelC}(b x) \, dx=\frac {1}{8} \, x^{8} \operatorname {C}\left (b x\right ) - \frac {\sqrt {\frac {1}{2}} {\left (-\left (105 i - 105\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) + \left (105 i + 105\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right ) + 28 \, {\left (\sqrt {\frac {1}{2}} \pi ^{3} b^{5} x^{5} - 15 \, \sqrt {\frac {1}{2}} \pi b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 4 \, {\left (\sqrt {\frac {1}{2}} \pi ^{4} b^{7} x^{7} - 35 \, \sqrt {\frac {1}{2}} \pi ^{2} b^{3} x^{3}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )}}{16 \, \pi ^{5} b^{8}} \] Input:

integrate(x^7*fresnel_cos(b*x),x, algorithm="maxima")
 

Output:

1/8*x^8*fresnel_cos(b*x) - 1/16*sqrt(1/2)*(-(105*I - 105)*(1/4)^(1/4)*pi*e 
rf(sqrt(1/2*I*pi)*b*x) + (105*I + 105)*(1/4)^(1/4)*pi*erf(sqrt(-1/2*I*pi)* 
b*x) + 28*(sqrt(1/2)*pi^3*b^5*x^5 - 15*sqrt(1/2)*pi*b*x)*cos(1/2*pi*b^2*x^ 
2) + 4*(sqrt(1/2)*pi^4*b^7*x^7 - 35*sqrt(1/2)*pi^2*b^3*x^3)*sin(1/2*pi*b^2 
*x^2))/(pi^5*b^8)
 

Giac [F]

\[ \int x^7 \operatorname {FresnelC}(b x) \, dx=\int { x^{7} \operatorname {C}\left (b x\right ) \,d x } \] Input:

integrate(x^7*fresnel_cos(b*x),x, algorithm="giac")
 

Output:

integrate(x^7*fresnel_cos(b*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^7 \operatorname {FresnelC}(b x) \, dx=\int x^7\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \] Input:

int(x^7*FresnelC(b*x),x)
 

Output:

int(x^7*FresnelC(b*x), x)
 

Reduce [F]

\[ \int x^7 \operatorname {FresnelC}(b x) \, dx=\int x^{7} \mathrm {FresnelC}\left (b x \right )d x \] Input:

int(x^7*FresnelC(b*x),x)
 

Output:

int(x^7*FresnelC(b*x),x)