3.117 \(\int C(b x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.00, size = 27, normalized size = 1.00 \[ x C(b x)-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b} \]

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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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\rm fresnelc}\left (b x\right ), x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm fresnelc}\left (b x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnelc(b*x), x)

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maple [A]  time = 0.01, size = 28, normalized size = 1.04 \[ \frac {b x \FresnelC \left (b x \right )-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm fresnelc}\left (b x\right )\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \mathrm {FresnelC}\left (b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x),x)

[Out]

int(FresnelC(b*x), x)

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sympy [B]  time = 0.70, size = 44, normalized size = 1.63 \[ \frac {x C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{4 \Gamma \left (\frac {5}{4}\right )} - \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{4 \pi b \Gamma \left (\frac {5}{4}\right )} \]

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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