3.67 \(\int \frac{\sin (\frac{1}{2} b^2 \pi x^2)}{S(b x)} \, dx\)

Optimal. Leaf size=9 \[ \frac{\log (S(b x))}{b} \]

[Out]

Log[FresnelS[b*x]]/b

________________________________________________________________________________________

Rubi [A]  time = 0.0147908, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {6440, 29} \[ \frac{\log (S(b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sin[(b^2*Pi*x^2)/2]/FresnelS[b*x],x]

[Out]

Log[FresnelS[b*x]]/b

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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.010098, size = 9, normalized size = 1. \[ \frac{\log (S(b x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[(b^2*Pi*x^2)/2]/FresnelS[b*x],x]

[Out]

Log[FresnelS[b*x]]/b

________________________________________________________________________________________

Maple [A]  time = 0.118, size = 10, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({\it FresnelS} \left ( bx \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(1/2*b^2*Pi*x^2)/FresnelS(b*x),x)

[Out]

ln(FresnelS(b*x))/b

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnels}\left (b x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/2*b^2*pi*x^2)/fresnels(b*x),x, algorithm="maxima")

[Out]

integrate(sin(1/2*pi*b^2*x^2)/fresnels(b*x), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnels}\left (b x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/2*b^2*pi*x^2)/fresnels(b*x),x, algorithm="fricas")

[Out]

integral(sin(1/2*pi*b^2*x^2)/fresnels(b*x), x)

________________________________________________________________________________________

Sympy [A]  time = 0.235245, size = 8, normalized size = 0.89 \begin{align*} \begin{cases} \frac{\log{\left (S\left (b x\right ) \right )}}{b} & \text{for}\: b \neq 0 \\\text{NaN} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/2*b**2*pi*x**2)/fresnels(b*x),x)

[Out]

Piecewise((log(fresnels(b*x))/b, Ne(b, 0)), (nan, True))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnels}\left (b x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/2*b^2*pi*x^2)/fresnels(b*x),x, algorithm="giac")

[Out]

integrate(sin(1/2*pi*b^2*x^2)/fresnels(b*x), x)