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

Optimal. Leaf size=27 \[ x \text{FresnelC}(b x)-\frac{\sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b} \]

[Out]

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

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Rubi [A]  time = 0.0047574, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6419} \[ x \text{FresnelC}(b x)-\frac{\sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b} \]

Antiderivative was successfully verified.

[In]

Int[FresnelC[b*x],x]

[Out]

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

Rule 6419

Int[FresnelC[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*FresnelC[a + b*x])/b, x] - Simp[Sin[(Pi*(a + b*
x)^2)/2]/(b*Pi), x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0021411, size = 27, normalized size = 1. \[ x \text{FresnelC}(b x)-\frac{\sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b} \]

Antiderivative was successfully verified.

[In]

Integrate[FresnelC[b*x],x]

[Out]

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

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Maple [A]  time = 0.051, size = 28, normalized size = 1. \begin{align*}{\frac{1}{b} \left ( bx{\it FresnelC} \left ( bx \right ) -{\frac{1}{\pi }\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnelc(b*x), 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 ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(fresnelc(b*x), x)

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Sympy [B]  time = 0.633405, size = 44, normalized size = 1.63 \begin{align*} \frac{x C\left (b x\right ) \Gamma \left (\frac{1}{4}\right )}{4 \Gamma \left (\frac{5}{4}\right )} - \frac{\sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac{1}{4}\right )}{4 \pi b \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x),x)

[Out]

x*fresnelc(b*x)*gamma(1/4)/(4*gamma(5/4)) - sin(pi*b**2*x**2/2)*gamma(1/4)/(4*pi*b*gamma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnelc(b*x), x)