3.8 \(\int S(b x) \, dx\)

Optimal. Leaf size=26 \[ \frac{\cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+x S(b x) \]

[Out]

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

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Rubi [A]  time = 0.0048741, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6418} \[ \frac{\cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+x S(b x) \]

Antiderivative was successfully verified.

[In]

Int[FresnelS[b*x],x]

[Out]

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

Rule 6418

Int[FresnelS[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*FresnelS[a + b*x])/b, x] + Simp[Cos[(Pi*(a + b*
x)^2)/2]/(b*Pi), x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0020637, size = 26, normalized size = 1. \[ \frac{\cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+x S(b x) \]

Antiderivative was successfully verified.

[In]

Integrate[FresnelS[b*x],x]

[Out]

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

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Maple [A]  time = 0.046, size = 27, normalized size = 1. \begin{align*}{\frac{1}{b} \left ( bx{\it FresnelS} \left ( bx \right ) +{\frac{1}{\pi }\cos \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)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnels(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(fresnels(b*x), x)

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Sympy [B]  time = 0.83294, size = 48, normalized size = 1.85 \begin{align*} \frac{3 x S\left (b x\right ) \Gamma \left (\frac{3}{4}\right )}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{3 \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac{3}{4}\right )}{4 \pi b \Gamma \left (\frac{7}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x),x)

[Out]

3*x*fresnels(b*x)*gamma(3/4)/(4*gamma(7/4)) + 3*cos(pi*b**2*x**2/2)*gamma(3/4)/(4*pi*b*gamma(7/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnels(b*x), x)