3.3 \(\int x^5 S(b x) \, dx\)

Optimal. Leaf size=99 \[ \frac{5 \text{FresnelC}(b x)}{2 \pi ^3 b^6}-\frac{5 x^3 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{6 \pi ^2 b^3}+\frac{x^5 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{6 \pi b}-\frac{5 x \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi ^3 b^5}+\frac{1}{6} x^6 S(b x) \]

[Out]

(-5*x*Cos[(b^2*Pi*x^2)/2])/(2*b^5*Pi^3) + (x^5*Cos[(b^2*Pi*x^2)/2])/(6*b*Pi) + (5*FresnelC[b*x])/(2*b^6*Pi^3)
+ (x^6*FresnelS[b*x])/6 - (5*x^3*Sin[(b^2*Pi*x^2)/2])/(6*b^3*Pi^2)

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Rubi [A]  time = 0.0643839, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6426, 3385, 3386, 3352} \[ \frac{5 \text{FresnelC}(b x)}{2 \pi ^3 b^6}-\frac{5 x^3 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{6 \pi ^2 b^3}+\frac{x^5 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{6 \pi b}-\frac{5 x \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi ^3 b^5}+\frac{1}{6} x^6 S(b x) \]

Antiderivative was successfully verified.

[In]

Int[x^5*FresnelS[b*x],x]

[Out]

(-5*x*Cos[(b^2*Pi*x^2)/2])/(2*b^5*Pi^3) + (x^5*Cos[(b^2*Pi*x^2)/2])/(6*b*Pi) + (5*FresnelC[b*x])/(2*b^6*Pi^3)
+ (x^6*FresnelS[b*x])/6 - (5*x^3*Sin[(b^2*Pi*x^2)/2])/(6*b^3*Pi^2)

Rule 6426

Int[FresnelS[(b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*FresnelS[b*x])/(d*(m + 1)), x] -
Dist[b/(d*(m + 1)), Int[(d*x)^(m + 1)*Sin[(Pi*b^2*x^2)/2], x], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]

Rule 3385

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] + Dist[(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 3386

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] - Dist[(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 3352

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

Rubi steps

\begin{align*} \int x^5 S(b x) \, dx &=\frac{1}{6} x^6 S(b x)-\frac{1}{6} b \int x^6 \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{6 b \pi }+\frac{1}{6} x^6 S(b x)-\frac{5 \int x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{6 b \pi }\\ &=\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{6 b \pi }+\frac{1}{6} x^6 S(b x)-\frac{5 x^3 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{6 b^3 \pi ^2}+\frac{5 \int x^2 \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{2 b^3 \pi ^2}\\ &=-\frac{5 x \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{2 b^5 \pi ^3}+\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{6 b \pi }+\frac{1}{6} x^6 S(b x)-\frac{5 x^3 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{6 b^3 \pi ^2}+\frac{5 \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{2 b^5 \pi ^3}\\ &=-\frac{5 x \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{2 b^5 \pi ^3}+\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{6 b \pi }+\frac{5 C(b x)}{2 b^6 \pi ^3}+\frac{1}{6} x^6 S(b x)-\frac{5 x^3 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{6 b^3 \pi ^2}\\ \end{align*}

Mathematica [A]  time = 0.0685961, size = 79, normalized size = 0.8 \[ \frac{\pi ^3 b^6 x^6 S(b x)-5 \pi b^3 x^3 \sin \left (\frac{1}{2} \pi b^2 x^2\right )+b x \left (\pi ^2 b^4 x^4-15\right ) \cos \left (\frac{1}{2} \pi b^2 x^2\right )+15 \text{FresnelC}(b x)}{6 \pi ^3 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*FresnelS[b*x],x]

[Out]

(b*x*(-15 + b^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2] + 15*FresnelC[b*x] + b^6*Pi^3*x^6*FresnelS[b*x] - 5*b^3*Pi*x^3*S
in[(b^2*Pi*x^2)/2])/(6*b^6*Pi^3)

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Maple [A]  time = 0.049, size = 96, normalized size = 1. \begin{align*}{\frac{1}{{b}^{6}} \left ({\frac{{b}^{6}{x}^{6}{\it FresnelS} \left ( bx \right ) }{6}}+{\frac{{b}^{5}{x}^{5}}{6\,\pi }\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }-{\frac{5}{6\,\pi } \left ({\frac{{x}^{3}{b}^{3}}{\pi }\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }-3\,{\frac{1}{\pi } \left ( -{\frac{bx\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}+{\frac{{\it FresnelC} \left ( bx \right ) }{\pi }} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*FresnelS(b*x),x)

[Out]

1/b^6*(1/6*b^6*x^6*FresnelS(b*x)+1/6/Pi*b^5*x^5*cos(1/2*b^2*Pi*x^2)-5/6/Pi*(1/Pi*b^3*x^3*sin(1/2*b^2*Pi*x^2)-3
/Pi*(-1/Pi*b*x*cos(1/2*b^2*Pi*x^2)+1/Pi*FresnelC(b*x))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5}{\rm fresnels}\left (b x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*fresnels(b*x),x, algorithm="maxima")

[Out]

integrate(x^5*fresnels(b*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{5}{\rm fresnels}\left (b x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*fresnels(b*x),x, algorithm="fricas")

[Out]

integral(x^5*fresnels(b*x), x)

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Sympy [A]  time = 1.04664, size = 53, normalized size = 0.54 \begin{align*} \frac{\pi b^{3} x^{9} \Gamma \left (\frac{3}{4}\right ) \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{3}{4}, \frac{9}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{13}{4} \end{matrix}\middle |{- \frac{\pi ^{2} b^{4} x^{4}}{16}} \right )}}{32 \Gamma \left (\frac{7}{4}\right ) \Gamma \left (\frac{13}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*fresnels(b*x),x)

[Out]

pi*b**3*x**9*gamma(3/4)*gamma(9/4)*hyper((3/4, 9/4), (3/2, 7/4, 13/4), -pi**2*b**4*x**4/16)/(32*gamma(7/4)*gam
ma(13/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5}{\rm fresnels}\left (b x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*fresnels(b*x),x, algorithm="giac")

[Out]

integrate(x^5*fresnels(b*x), x)