3.43 \(\int \frac{S(b x)^2}{x^5} \, dx\)

Optimal. Leaf size=127 \[ -\frac{b S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{6 x^3}-\frac{\pi b^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{6 x}-\frac{1}{12} \pi ^2 b^4 S(b x)^2+\frac{1}{12} \pi b^4 \text{Si}\left (b^2 \pi x^2\right )-\frac{b^2}{24 x^2}+\frac{b^2 \cos \left (\pi b^2 x^2\right )}{24 x^2}-\frac{S(b x)^2}{4 x^4} \]

[Out]

-b^2/(24*x^2) + (b^2*Cos[b^2*Pi*x^2])/(24*x^2) - (b^3*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(6*x) - (b^4*Pi^2*
FresnelS[b*x]^2)/12 - FresnelS[b*x]^2/(4*x^4) - (b*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(6*x^3) + (b^4*Pi*SinInt
egral[b^2*Pi*x^2])/12

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Rubi [A]  time = 0.143154, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {6430, 6456, 6464, 6440, 30, 3375, 3380, 3297, 3299} \[ -\frac{b S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{6 x^3}-\frac{\pi b^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{6 x}-\frac{1}{12} \pi ^2 b^4 S(b x)^2+\frac{1}{12} \pi b^4 \text{Si}\left (b^2 \pi x^2\right )-\frac{b^2}{24 x^2}+\frac{b^2 \cos \left (\pi b^2 x^2\right )}{24 x^2}-\frac{S(b x)^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[FresnelS[b*x]^2/x^5,x]

[Out]

-b^2/(24*x^2) + (b^2*Cos[b^2*Pi*x^2])/(24*x^2) - (b^3*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(6*x) - (b^4*Pi^2*
FresnelS[b*x]^2)/12 - FresnelS[b*x]^2/(4*x^4) - (b*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(6*x^3) + (b^4*Pi*SinInt
egral[b^2*Pi*x^2])/12

Rule 6430

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

Rule 6456

Int[FresnelS[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[(x^(m + 1)*Sin[d*x^2]*FresnelS[b*x])/(
m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*Cos[d*x^2]*FresnelS[b*x], x], x] + Dist[d/(Pi*b*(m + 1)), Int
[x^(m + 1)*Cos[2*d*x^2], x], x] - Simp[(d*x^(m + 2))/(Pi*b*(m + 1)*(m + 2)), x]) /; FreeQ[{b, d}, x] && EqQ[d^
2, (Pi^2*b^4)/4] && ILtQ[m, -2]

Rule 6464

Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m + 1)*Cos[d*x^2]*FresnelS[b*x])/(
m + 1), x] + (Dist[(2*d)/(m + 1), Int[x^(m + 2)*Sin[d*x^2]*FresnelS[b*x], x], x] - Dist[d/(Pi*b*(m + 1)), Int[
x^(m + 1)*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4] && ILtQ[m, -1]

Rule 6440

Int[FresnelS[(b_.)*(x_)]^(n_.)*Sin[(d_.)*(x_)^2], x_Symbol] :> Dist[(Pi*b)/(2*d), Subst[Int[x^n, x], x, Fresne
lS[b*x]], x] /; FreeQ[{b, d, n}, x] && EqQ[d^2, (Pi^2*b^4)/4]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 3375

Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

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 3297

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

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{S(b x)^2}{x^5} \, dx &=-\frac{S(b x)^2}{4 x^4}+\frac{1}{2} b \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=-\frac{b^2}{24 x^2}-\frac{S(b x)^2}{4 x^4}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{6 x^3}-\frac{1}{12} b^2 \int \frac{\cos \left (b^2 \pi x^2\right )}{x^3} \, dx+\frac{1}{6} \left (b^3 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^2} \, dx\\ &=-\frac{b^2}{24 x^2}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{6 x}-\frac{S(b x)^2}{4 x^4}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{6 x^3}-\frac{1}{24} b^2 \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac{1}{12} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x} \, dx-\frac{1}{6} \left (b^5 \pi ^2\right ) \int S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac{b^2}{24 x^2}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{6 x}-\frac{S(b x)^2}{4 x^4}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{6 x^3}+\frac{1}{24} b^4 \pi \text{Si}\left (b^2 \pi x^2\right )+\frac{1}{24} \left (b^4 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac{1}{6} \left (b^4 \pi ^2\right ) \operatorname{Subst}(\int x \, dx,x,S(b x))\\ &=-\frac{b^2}{24 x^2}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{6 x}-\frac{1}{12} b^4 \pi ^2 S(b x)^2-\frac{S(b x)^2}{4 x^4}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{6 x^3}+\frac{1}{12} b^4 \pi \text{Si}\left (b^2 \pi x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0060233, size = 127, normalized size = 1. \[ -\frac{b S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{6 x^3}-\frac{\pi b^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{6 x}-\frac{1}{12} \pi ^2 b^4 S(b x)^2+\frac{1}{12} \pi b^4 \text{Si}\left (b^2 \pi x^2\right )-\frac{b^2}{24 x^2}+\frac{b^2 \cos \left (\pi b^2 x^2\right )}{24 x^2}-\frac{S(b x)^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[FresnelS[b*x]^2/x^5,x]

[Out]

-b^2/(24*x^2) + (b^2*Cos[b^2*Pi*x^2])/(24*x^2) - (b^3*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(6*x) - (b^4*Pi^2*
FresnelS[b*x]^2)/12 - FresnelS[b*x]^2/(4*x^4) - (b*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(6*x^3) + (b^4*Pi*SinInt
egral[b^2*Pi*x^2])/12

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it FresnelS} \left ( bx \right ) \right ) ^{2}}{{x}^{5}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(b*x)^2/x^5,x)

[Out]

int(FresnelS(b*x)^2/x^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right )^{2}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^5,x, algorithm="maxima")

[Out]

integrate(fresnels(b*x)^2/x^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnels}\left (b x\right )^{2}}{x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^5,x, algorithm="fricas")

[Out]

integral(fresnels(b*x)^2/x^5, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{S^{2}\left (b x\right )}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)**2/x**5,x)

[Out]

Integral(fresnels(b*x)**2/x**5, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right )^{2}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^5,x, algorithm="giac")

[Out]

integrate(fresnels(b*x)^2/x^5, x)