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

Optimal. Leaf size=27 \[ \frac{1}{2} b \text{Si}\left (\frac{1}{2} b^2 \pi x^2\right )-\frac{S(b x)}{x} \]

[Out]

-(FresnelS[b*x]/x) + (b*SinIntegral[(b^2*Pi*x^2)/2])/2

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Rubi [A]  time = 0.0215755, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6426, 3375} \[ \frac{1}{2} b \text{Si}\left (\frac{1}{2} b^2 \pi x^2\right )-\frac{S(b x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(FresnelS[b*x]/x) + (b*SinIntegral[(b^2*Pi*x^2)/2])/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 3375

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

Rubi steps

\begin{align*} \int \frac{S(b x)}{x^2} \, dx &=-\frac{S(b x)}{x}+b \int \frac{\sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx\\ &=-\frac{S(b x)}{x}+\frac{1}{2} b \text{Si}\left (\frac{1}{2} b^2 \pi x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.010818, size = 27, normalized size = 1. \[ \frac{1}{2} b \text{Si}\left (\frac{1}{2} b^2 \pi x^2\right )-\frac{S(b x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(FresnelS[b*x]/x) + (b*SinIntegral[(b^2*Pi*x^2)/2])/2

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Maple [A]  time = 0.049, size = 28, normalized size = 1. \begin{align*} b \left ( -{\frac{{\it FresnelS} \left ( bx \right ) }{bx}}+{\frac{1}{2}{\it Si} \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*(-FresnelS(b*x)/b/x+1/2*Si(1/2*b^2*Pi*x^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnels(b*x)/x^2, 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 )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(fresnels(b*x)/x^2, x)

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Sympy [B]  time = 0.544513, size = 42, normalized size = 1.56 \begin{align*} \frac{\pi b^{3} x^{2} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{3}{2}, \frac{3}{2}, \frac{7}{4} \end{matrix}\middle |{- \frac{\pi ^{2} b^{4} x^{4}}{16}} \right )}}{16 \Gamma \left (\frac{7}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

pi*b**3*x**2*gamma(3/4)*hyper((1/2, 3/4), (3/2, 3/2, 7/4), -pi**2*b**4*x**4/16)/(16*gamma(7/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnels(b*x)/x^2, x)