3.3.8 \(\int \text {FresnelC}(b x) \sin (\frac {1}{2} b^2 \pi x^2) \, dx\) [208]

Optimal. Leaf size=80 \[ \frac {\text {FresnelC}(b x) S(b x)}{2 b}+\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right ) \]

[Out]

1/2*FresnelC(b*x)*FresnelS(b*x)/b+1/8*I*b*x^2*hypergeom([1, 1],[3/2, 2],-1/2*I*b^2*Pi*x^2)-1/8*I*b*x^2*hyperge
om([1, 1],[3/2, 2],1/2*I*b^2*Pi*x^2)

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Rubi [A]
time = 0.01, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6582} \begin {gather*} \frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )+\frac {\text {FresnelC}(b x) S(b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(FresnelC[b*x]*FresnelS[b*x])/(2*b) + (I/8)*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-1/2*I)*b^2*Pi*x^2] - (
I/8)*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2]

Rule 6582

Int[FresnelC[(b_.)*(x_)]*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[b*Pi*FresnelC[b*x]*(FresnelS[b*x]/(4*d)), x] + (
Simp[(1/8)*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-I)*d*x^2], x] - Simp[(1/8)*I*b*x^2*HypergeometricPFQ[
{1, 1}, {3/2, 2}, I*d*x^2], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4]

Rubi steps

\begin {align*} \int C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx &=\frac {C(b x) S(b x)}{2 b}+\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )\\ \end {align*}

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Mathematica [F]
time = 0.01, size = 0, normalized size = 0.00 \begin {gather*} \int \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
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(fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="maxima")

[Out]

integrate(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2), x)

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Fricas [F]
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(fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="fricas")

[Out]

integral(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)*sin(1/2*b**2*pi*x**2),x)

[Out]

Integral(sin(pi*b**2*x**2/2)*fresnelc(b*x), x)

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Giac [F]
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(fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {FresnelC}\left (b\,x\right )\,\sin \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(FresnelC(b*x)*sin((Pi*b^2*x^2)/2),x)

[Out]

int(FresnelC(b*x)*sin((Pi*b^2*x^2)/2), x)

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