3.111 \(\int x^6 \text{FresnelC}(b x) \, dx\)

Optimal. Leaf size=109 \[ -\frac{x^6 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi b}+\frac{24 x^2 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}-\frac{6 x^4 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac{48 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac{1}{7} x^7 \text{FresnelC}(b x) \]

[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)

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Rubi [A]  time = 0.114325, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6427, 3380, 3296, 2638} \[ -\frac{x^6 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi b}+\frac{24 x^2 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}-\frac{6 x^4 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac{48 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac{1}{7} x^7 \text{FresnelC}(b x) \]

Antiderivative was successfully verified.

[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 6427

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*b^2*x^2)/2], x], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]

Rule 3380

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 3296

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 2638

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

Rubi steps

\begin{align*} \int x^6 C(b x) \, dx &=\frac{1}{7} x^7 C(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 C(b x)-\frac{1}{14} b \operatorname{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 C(b x)-\frac{x^6 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac{3 \operatorname{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 C(b x)-\frac{x^6 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac{12 \operatorname{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 C(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 \operatorname{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 C(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]  time = 0.0559025, size = 83, normalized size = 0.76 \[ -\frac{x^2 \left (\pi ^2 b^4 x^4-24\right ) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}-\frac{6 \left (\pi ^2 b^4 x^4-8\right ) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac{1}{7} x^7 \text{FresnelC}(b x) \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.049, size = 107, normalized size = 1. \begin{align*}{\frac{1}{{b}^{7}} \left ({\frac{{b}^{7}{x}^{7}{\it FresnelC} \left ( bx \right ) }{7}}-{\frac{{b}^{6}{x}^{6}}{7\,\pi }\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+{\frac{6}{7\,\pi } \left ( -{\frac{{x}^{4}{b}^{4}}{\pi }\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+4\,{\frac{1}{\pi } \left ({\frac{{b}^{2}{x}^{2}\sin \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}+2\,{\frac{\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{{\pi }^{2}}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^7*(1/7*b^7*x^7*FresnelC(b*x)-1/7/Pi*b^6*x^6*sin(1/2*b^2*Pi*x^2)+6/7/Pi*(-1/Pi*b^4*x^4*cos(1/2*b^2*Pi*x^2)+
4/Pi*(1/Pi*b^2*x^2*sin(1/2*b^2*Pi*x^2)+2/Pi^2*cos(1/2*b^2*Pi*x^2))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{6}{\rm fresnelc}\left (b x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^6*fresnelc(b*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{6}{\rm fresnelc}\left (b x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^6*fresnelc(b*x), x)

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Sympy [A]  time = 2.36106, size = 153, normalized size = 1.4 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{6}{\rm fresnelc}\left (b x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^6*fresnelc(b*x), x)