3.41 \(\int \frac{S(b x)^2}{x^3} \, dx\)

Optimal. Leaf size=38 \[ b \text{Unintegrable}\left (\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x^2},x\right )-\frac{S(b x)^2}{2 x^2} \]

[Out]

-FresnelS[b*x]^2/(2*x^2) + b*Unintegrable[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x]

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Rubi [A]  time = 0.0383717, 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 = {} \[ \int \frac{S(b x)^2}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[FresnelS[b*x]^2/x^3,x]

[Out]

-FresnelS[b*x]^2/(2*x^2) + b*Defer[Int][(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x]

Rubi steps

\begin{align*} \int \frac{S(b x)^2}{x^3} \, dx &=-\frac{S(b x)^2}{2 x^2}+b \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0198748, size = 0, normalized size = 0. \[ \int \frac{S(b x)^2}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[FresnelS[b*x]^2/x^3,x]

[Out]

Integrate[FresnelS[b*x]^2/x^3, x]

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Maple [A]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it FresnelS} \left ( bx \right ) \right ) ^{2}}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(b*x)^2/x^3,x)

[Out]

int(FresnelS(b*x)^2/x^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right )^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^3,x, algorithm="maxima")

[Out]

integrate(fresnels(b*x)^2/x^3, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnels}\left (b x\right )^{2}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^3,x, algorithm="fricas")

[Out]

integral(fresnels(b*x)^2/x^3, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{S^{2}\left (b x\right )}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)**2/x**3,x)

[Out]

Integral(fresnels(b*x)**2/x**3, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right )^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^3,x, algorithm="giac")

[Out]

integrate(fresnels(b*x)^2/x^3, x)