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

Optimal. Leaf size=17 \[ \frac{\text{FresnelC}(b x)^{n+1}}{b (n+1)} \]

[Out]

FresnelC[b*x]^(1 + n)/(b*(1 + n))

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Rubi [A]  time = 0.0187874, antiderivative size = 17, 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} \[ \frac{\text{FresnelC}(b x)^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

FresnelC[b*x]^(1 + n)/(b*(1 + n))

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)^n \, dx &=\frac{\operatorname{Subst}\left (\int x^n \, dx,x,C(b x)\right )}{b}\\ &=\frac{C(b x)^{1+n}}{b (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0102103, size = 17, normalized size = 1. \[ \frac{\text{FresnelC}(b x)^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

FresnelC[b*x]^(1 + n)/(b*(1 + n))

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Maple [A]  time = 0.047, size = 18, normalized size = 1.1 \begin{align*}{\frac{ \left ({\it FresnelC} \left ( bx \right ) \right ) ^{1+n}}{b \left ( 1+n \right ) }} \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)^n,x)

[Out]

FresnelC(b*x)^(1+n)/b/(1+n)

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

[Out]

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

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

[Out]

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

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Sympy [A]  time = 4.59086, size = 34, normalized size = 2. \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: b = 0 \wedge n = -1 \\0^{n} x & \text{for}\: b = 0 \\\frac{\log{\left (C\left (b x\right ) \right )}}{b} & \text{for}\: n = -1 \\\frac{C\left (b x\right ) C^{n}\left (b x\right )}{b n + b} & \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)**n,x)

[Out]

Piecewise((zoo*x, Eq(b, 0) & Eq(n, -1)), (0**n*x, Eq(b, 0)), (log(fresnelc(b*x))/b, Eq(n, -1)), (fresnelc(b*x)
*fresnelc(b*x)**n/(b*n + b), True))

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

[Out]

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