\(\int x^6 \operatorname {FresnelC}(b x) \, dx\) [111]

Optimal result

Integrand size = 8, antiderivative size = 109 \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}+\frac {1}{7} x^7 \operatorname {FresnelC}(b x)+\frac {24 x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}-\frac {x^6 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b \pi } \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6562, 3461, 3377, 2718} \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=-\frac {x^6 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi b}+\frac {48 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac {24 x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}-\frac {6 x^4 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac {1}{7} x^7 \operatorname {FresnelC}(b x) \]

[In]

Int[x^6*FresnelC[b*x],x]

[Out]

(48*Cos[(b^2*Pi*x^2)/2])/(7*b^7*Pi^4) - (6*x^4*Cos[(b^2*Pi*x^2)/2])/(7*b^3*Pi^2) + (x^7*FresnelC[b*x])/7 + (24
*x^2*Sin[(b^2*Pi*x^2)/2])/(7*b^5*Pi^3) - (x^6*Sin[(b^2*Pi*x^2)/2])/(7*b*Pi)

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3461

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(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[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 6562

Int[FresnelC[(b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*(FresnelC[b*x]/(d*(m + 1))), x] -
Dist[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]

Rubi steps \begin{align*} \text {integral}& = \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 \\ & = \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {1}{14} b \text {Subst}\left (\int x^3 \cos \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {x^6 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac {3 \text {Subst}\left (\int x^2 \sin \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{7 b \pi } \\ & = -\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 {x^6 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac {12 \text {Subst}\left (\int x \cos \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{7 b^3 \pi ^2} \\ & = -\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 }-\frac {24 \text {Subst}\left (\int \sin \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{7 b^5 \pi ^3} \\ & = \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 } \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.76 \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=-\frac {6 \left (-8+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}+\frac {1}{7} x^7 \operatorname {FresnelC}(b x)-\frac {x^2 \left (-24+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3} \]

[In]

Integrate[x^6*FresnelC[b*x],x]

[Out]

(-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)

Maple [C] (verified)

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	\(\frac {x^{7} \operatorname {FresnelC}\left (b x \right )}{7}-\frac {b \left (\frac {x^{6} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {6 \left (-\frac {x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {\frac {4 x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{4} \pi ^{2}}}{b^{2} \pi }\right )}{b^{2} \pi }\right )}{7}\)	\(112\)

[In]

int(x^6*FresnelC(b*x),x,method=_RETURNVERBOSE)

[Out]

1/8*b*x^8*hypergeom([1/4,2],[1/2,5/4,3],-1/16*x^4*Pi^2*b^4)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.72 \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=\frac {\pi ^{4} b^{7} x^{7} \operatorname {C}\left (b x\right ) - 6 \, {\left (\pi ^{2} b^{4} x^{4} - 8\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{3} b^{6} x^{6} - 24 \, \pi b^{2} x^{2}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{7 \, \pi ^{4} b^{7}} \]

[In]

integrate(x^6*fresnel_cos(b*x),x, algorithm="fricas")

[Out]

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)

Sympy [A] (verification not implemented)

Time = 1.17 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.40 \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=\frac {x^{7} C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{28 \Gamma \left (\frac {5}{4}\right )} - \frac {x^{6} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{28 \pi b \Gamma \left (\frac {5}{4}\right )} - \frac {3 x^{4} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{14 \pi ^{2} b^{3} \Gamma \left (\frac {5}{4}\right )} + \frac {6 x^{2} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{7 \pi ^{3} b^{5} \Gamma \left (\frac {5}{4}\right )} + \frac {12 \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{7 \pi ^{4} b^{7} \Gamma \left (\frac {5}{4}\right )} \]

[In]

integrate(x**6*fresnelc(b*x),x)

[Out]

x**7*fresnelc(b*x)*gamma(1/4)/(28*gamma(5/4)) - x**6*sin(pi*b**2*x**2/2)*gamma(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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.68 \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=\frac {1}{7} \, x^{7} \operatorname {C}\left (b x\right ) - \frac {6 \, {\left (\pi ^{2} b^{4} x^{4} - 8\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{3} b^{6} x^{6} - 24 \, \pi b^{2} x^{2}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{7 \, \pi ^{4} b^{7}} \]

[In]

integrate(x^6*fresnel_cos(b*x),x, algorithm="maxima")

[Out]

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)

Giac [F]

\[ \int x^6 \operatorname {FresnelC}(b x) \, dx=\int { x^{6} \operatorname {C}\left (b x\right ) \,d x } \]

[In]

integrate(x^6*fresnel_cos(b*x),x, algorithm="giac")

[Out]

integrate(x^6*fresnel_cos(b*x), x)

Mupad [F(-1)]

Timed out. \[ \int x^6 \operatorname {FresnelC}(b x) \, dx=\int x^6\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]

[In]

int(x^6*FresnelC(b*x),x)

[Out]

int(x^6*FresnelC(b*x), x)