3.7 \(\int x S(b x) \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6426, 3385, 3352} \[ -\frac {\text {FresnelC}(b x)}{2 \pi b^2}+\frac {x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 \pi b}+\frac {1}{2} x^2 S(b x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3352

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

Rule 3385

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

Rule 6426

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 49, normalized size = 1.00 \[ -\frac {C(b x)}{2 \pi b^2}+\frac {x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 \pi b}+\frac {1}{2} x^2 S(b x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x {\rm fresnels}\left (b x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*fresnels(b*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*fresnels(b*x), x)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 44, normalized size = 0.90 \[ \frac {\frac {b^{2} x^{2} \mathrm {S}\left (b x \right )}{2}+\frac {b x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 \pi }-\frac {\FresnelC \left (b x \right )}{2 \pi }}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*fresnels(b*x), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\mathrm {FresnelS}\left (b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*FresnelS(b*x), x)

________________________________________________________________________________________

sympy [A]  time = 0.52, size = 53, normalized size = 1.08 \[ \frac {\pi b^{3} x^{5} \Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {3}{2}, \frac {7}{4}, \frac {9}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{32 \Gamma \left (\frac {7}{4}\right ) \Gamma \left (\frac {9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

pi*b**3*x**5*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -pi**2*b**4*x**4/16)/(32*gamma(7/4)*gamm
a(9/4))

________________________________________________________________________________________