3.2.74 \(\int \cos (\frac {1}{2} b^2 \pi x^2) \text {FresnelC}(b x)^2 \, dx\) [174]

Optimal. Leaf size=13 \[ \frac {\text {FresnelC}(b x)^3}{3 b} \]

[Out]

1/3*FresnelC(b*x)^3/b

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Rubi [A]
time = 0.01, antiderivative size = 13, 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 = {6576, 30} \begin {gather*} \frac {\text {FresnelC}(b x)^3}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x]^2,x]

[Out]

FresnelC[b*x]^3/(3*b)

Rule 30

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

Rule 6576

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

Rubi steps

\begin {align*} \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)^2 \, dx &=\frac {\text {Subst}\left (\int x^2 \, dx,x,C(b x)\right )}{b}\\ &=\frac {C(b x)^3}{3 b}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {\text {FresnelC}(b x)^3}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x]^2,x]

[Out]

FresnelC[b*x]^3/(3*b)

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Maple [A]
time = 0.17, size = 12, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {\FresnelC \left (b x \right )^{3}}{3 b}\)	\(12\)
default	\(\frac {\FresnelC \left (b x \right )^{3}}{3 b}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/3*FresnelC(b*x)^3/b

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Maxima [A]
time = 0.26, size = 11, normalized size = 0.85 \begin {gather*} \frac {\operatorname {C}\left (b x\right )^{3}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*fresnel_cos(b*x)^3/b

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Fricas [A]
time = 0.35, size = 11, normalized size = 0.85 \begin {gather*} \frac {\operatorname {C}\left (b x\right )^{3}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*fresnel_cos(b*x)^3/b

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Sympy [A]
time = 0.20, size = 10, normalized size = 0.77 \begin {gather*} \begin {cases} \frac {C^{3}\left (b x\right )}{3 b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b**2*pi*x**2)*fresnelc(b*x)**2,x)

[Out]

Piecewise((fresnelc(b*x)**3/(3*b), Ne(b, 0)), (0, 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(cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x)^2,x, algorithm="giac")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int {\mathrm {FresnelC}\left (b\,x\right )}^2\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)^2*cos((Pi*b^2*x^2)/2),x)

[Out]

int(FresnelC(b*x)^2*cos((Pi*b^2*x^2)/2), x)

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