3.116 \(\int x \text{FresnelC}(b x) \, dx\)

Optimal. Leaf size=49 \[ \frac{S(b x)}{2 \pi b^2}-\frac{x \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi b}+\frac{1}{2} x^2 \text{FresnelC}(b x) \]

[Out]

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

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Rubi [A]  time = 0.0258736, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6427, 3386, 3351} \[ \frac{S(b x)}{2 \pi b^2}-\frac{x \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi b}+\frac{1}{2} x^2 \text{FresnelC}(b x) \]

Antiderivative was successfully verified.

[In]

Int[x*FresnelC[b*x],x]

[Out]

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

Rule 6427

Int[FresnelC[(b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*FresnelC[b*x])/(d*(m + 1)), x] -
Dist[b/(d*(m + 1)), Int[(d*x)^(m + 1)*Cos[(Pi*b^2*x^2)/2], x], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]

Rule 3386

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Sin[c + d*
x^n])/(d*n), x] - Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x
] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

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

Mathematica [A]  time = 0.0117981, size = 49, normalized size = 1. \[ \frac{S(b x)}{2 \pi b^2}-\frac{x \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi b}+\frac{1}{2} x^2 \text{FresnelC}(b x) \]

Antiderivative was successfully verified.

[In]

Integrate[x*FresnelC[b*x],x]

[Out]

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

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Maple [A]  time = 0.048, size = 44, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{{b}^{2}{x}^{2}{\it FresnelC} \left ( bx \right ) }{2}}-{\frac{bx}{2\,\pi }\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+{\frac{{\it FresnelS} \left ( bx \right ) }{2\,\pi }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*FresnelC(b*x),x)

[Out]

1/b^2*(1/2*b^2*x^2*FresnelC(b*x)-1/2/Pi*b*x*sin(1/2*b^2*Pi*x^2)+1/2/Pi*FresnelS(b*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*fresnelc(b*x),x, algorithm="maxima")

[Out]

integrate(x*fresnelc(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*fresnelc(b*x),x, algorithm="fricas")

[Out]

integral(x*fresnelc(b*x), x)

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Sympy [A]  time = 0.525939, size = 49, normalized size = 1. \begin{align*} \frac{b x^{3} \Gamma \left (\frac{1}{4}\right ) \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle |{- \frac{\pi ^{2} b^{4} x^{4}}{16}} \right )}}{16 \Gamma \left (\frac{5}{4}\right ) \Gamma \left (\frac{7}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*fresnelc(b*x),x)

[Out]

b*x**3*gamma(1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -pi**2*b**4*x**4/16)/(16*gamma(5/4)*gamma(7/4)
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*fresnelc(b*x),x, algorithm="giac")

[Out]

integrate(x*fresnelc(b*x), x)