3.1.67 \(\int \frac {\sin (\frac {1}{2} b^2 \pi x^2)}{S(b x)} \, dx\) [67]

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

[Out]

ln(FresnelS(b*x))/b

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Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 = {6575, 29} \begin {gather*} \frac {\log (S(b x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[FresnelS[b*x]]/b

Rule 29

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

Rule 6575

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^2/4)*b^4]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 9, normalized size = 1.00 \begin {gather*} \frac {\log (S(b x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[FresnelS[b*x]]/b

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Maple [A]
time = 0.12, size = 10, normalized size = 1.11

method	result	size
derivativedivides	\(\frac {\ln \left (\mathrm {S}\left (b x \right )\right )}{b}\)	\(10\)
default	\(\frac {\ln \left (\mathrm {S}\left (b x \right )\right )}{b}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(FresnelS(b*x))/b

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Maxima [A]
time = 0.25, size = 9, normalized size = 1.00 \begin {gather*} \frac {\log \left (\operatorname {S}\left (b x\right )\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(fresnel_sin(b*x))/b

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Fricas [A]
time = 0.33, size = 9, normalized size = 1.00 \begin {gather*} \frac {\log \left (\operatorname {S}\left (b x\right )\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(fresnel_sin(b*x))/b

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Sympy [A]
time = 0.10, size = 8, normalized size = 0.89 \begin {gather*} \begin {cases} \frac {\log {\left (S\left (b x\right ) \right )}}{b} & \text {for}\: b \neq 0 \\\text {NaN} & \text {otherwise} \end {cases} \end {gather*}

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))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.11 \begin {gather*} \int \frac {\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{\mathrm {FresnelS}\left (b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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