3.1.37 \(\int x S(b x)^2 \, dx\) [37]

Optimal. Leaf size=143 \[ \frac {\cos \left (b^2 \pi x^2\right )}{4 b^2 \pi ^2}+\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b \pi }-\frac {\text {FresnelC}(b x) S(b x)}{2 b^2 \pi }+\frac {1}{2} x^2 S(b x)^2+\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi } \]

[Out]

1/4*cos(b^2*Pi*x^2)/b^2/Pi^2+x*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b/Pi-1/2*FresnelC(b*x)*FresnelS(b*x)/b^2/Pi+1
/2*x^2*FresnelS(b*x)^2+1/8*I*x^2*hypergeom([1, 1],[3/2, 2],-1/2*I*b^2*Pi*x^2)/Pi-1/8*I*x^2*hypergeom([1, 1],[3
/2, 2],1/2*I*b^2*Pi*x^2)/Pi

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Rubi [A]
time = 0.06, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6565, 6589, 6581, 3460, 2718} \begin {gather*} \frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {\text {FresnelC}(b x) S(b x)}{2 \pi b^2}+\frac {x S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^2}+\frac {1}{2} x^2 S(b x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[b^2*Pi*x^2]/(4*b^2*Pi^2) + (x*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b*Pi) - (FresnelC[b*x]*FresnelS[b*x])/(2
*b^2*Pi) + (x^2*FresnelS[b*x]^2)/2 + ((I/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-1/2*I)*b^2*Pi*x^2])/Pi -
((I/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2])/Pi

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3460

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 6565

Int[FresnelS[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(FresnelS[b*x]^2/(m + 1)), x] - Dist[2*(b/(
m + 1)), Int[x^(m + 1)*Sin[(Pi/2)*b^2*x^2]*FresnelS[b*x], x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]

Rule 6581

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

Rule 6589

Int[FresnelS[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[(-x^(m - 1))*Cos[d*x^2]*(FresnelS[b*x]
/(2*d)), x] + (Dist[(m - 1)/(2*d), Int[x^(m - 2)*Cos[d*x^2]*FresnelS[b*x], x], x] + Dist[1/(2*b*Pi), Int[x^(m
- 1)*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IGtQ[m, 1]

Rubi steps

\begin {align*} \int x S(b x)^2 \, dx &=\frac {1}{2} x^2 S(b x)^2-b \int x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b \pi }+\frac {1}{2} x^2 S(b x)^2-\frac {\int x \sin \left (b^2 \pi x^2\right ) \, dx}{2 \pi }-\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b \pi }\\ &=\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b \pi }-\frac {C(b x) S(b x)}{2 b^2 \pi }+\frac {1}{2} x^2 S(b x)^2+\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {\text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 \pi }\\ &=\frac {\cos \left (b^2 \pi x^2\right )}{4 b^2 \pi ^2}+\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b \pi }-\frac {C(b x) S(b x)}{2 b^2 \pi }+\frac {1}{2} x^2 S(b x)^2+\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }\\ \end {align*}

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Mathematica [F]
time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int x S(b x)^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Integrate[x*FresnelS[b*x]^2, x]

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int x \mathrm {S}\left (b x \right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*FresnelS(b*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(x*fresnel_sin(b*x)^2,x, algorithm="maxima")

[Out]

integrate(x*fresnel_sin(b*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(x*fresnel_sin(b*x)^2,x, algorithm="fricas")

[Out]

integral(x*fresnel_sin(b*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*fresnels(b*x)**2, 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(x*fresnel_sin(b*x)^2,x, algorithm="giac")

[Out]

integrate(x*fresnel_sin(b*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*FresnelS(b*x)^2, x)

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