3.90 \(\int \cos (\frac{1}{2} b^2 \pi x^2) S(b x)^n \, dx\)

Optimal. Leaf size=21 \[ \text{Unintegrable}\left (\cos \left (\frac{1}{2} \pi b^2 x^2\right ) S(b x)^n,x\right ) \]

[Out]

Unintegrable[Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x]^n, x]

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Rubi [A]  time = 0.0146736, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)^n \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0730368, size = 0, normalized size = 0. \[ \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)^n \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.062, size = 0, normalized size = 0. \begin{align*} \int \cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) \left ({\it FresnelS} \left ( bx \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm fresnels}\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)*fresnels(b*x)^n,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\rm fresnels}\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)*fresnels(b*x)^n,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S^{n}\left (b x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(pi*b**2*x**2/2)*fresnels(b*x)**n, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm fresnels}\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)*fresnels(b*x)^n,x, algorithm="giac")

[Out]

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