3.1.90 \(\int \cos (\frac {1}{2} b^2 \pi x^2) S(b x)^n \, dx\) [90]

Optimal. Leaf size=22 \[ \text {Int}\left (\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)^n,x\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)^n \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)^n \, dx \end {gather*}

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 = 0.23, size = 0, normalized size = 0.00 \[\int \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )^{n}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
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_sin(b*x)^n,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S^{n}\left (b x\right )\, dx \end {gather*}

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 = 0.00, 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_sin(b*x)^n,x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int {\mathrm {FresnelS}\left (b\,x\right )}^n\,\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(FresnelS(b*x)^n*cos((Pi*b^2*x^2)/2),x)

[Out]

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

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