3.174 \(\int \cos (\frac{1}{2} b^2 \pi x^2) \text{FresnelC}(b x)^2 \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0148672, antiderivative size = 13, 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 = {6441, 30} \[ \frac{\text{FresnelC}(b x)^3}{3 b} \]

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 6441

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*b^4)/4]

Rule 30

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

Rubi steps

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

Mathematica [A]  time = 0.0087997, size = 13, normalized size = 1. \[ \frac{\text{FresnelC}(b x)^3}{3 b} \]

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)

________________________________________________________________________________________

Maple [A]  time = 0.044, size = 12, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it FresnelC} \left ( bx \right ) \right ) ^{3}}{3\,b}} \end{align*}

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)

[Out]

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

________________________________________________________________________________________

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

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, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

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, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.41009, size = 10, normalized size = 0.77 \begin{align*} \begin{cases} \frac{C^{3}\left (b x\right )}{3 b} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

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))

________________________________________________________________________________________

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

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, algorithm="giac")

[Out]

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