3.75 \(\int x^4 S(b x) \sin (\frac{1}{2} b^2 \pi x^2) \, dx\)

Optimal. Leaf size=120 \[ \frac{3 x S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{x^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{3 S(b x)^2}{2 \pi ^2 b^5}-\frac{3 x^2}{4 \pi ^2 b^3}+\frac{\sin \left (\pi b^2 x^2\right )}{\pi ^3 b^5}-\frac{x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]

[Out]

(-3*x^2)/(4*b^3*Pi^2) - (x^2*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (x^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^2*Pi)
- (3*FresnelS[b*x]^2)/(2*b^5*Pi^2) + (3*x*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^4*Pi^2) + Sin[b^2*Pi*x^2]/(b^5
*Pi^3)

________________________________________________________________________________________

Rubi [A]  time = 0.118406, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6454, 6462, 3379, 2634, 6440, 30, 3296, 2637} \[ \frac{3 x S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{x^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{3 S(b x)^2}{2 \pi ^2 b^5}-\frac{3 x^2}{4 \pi ^2 b^3}+\frac{\sin \left (\pi b^2 x^2\right )}{\pi ^3 b^5}-\frac{x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^4*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2],x]

[Out]

(-3*x^2)/(4*b^3*Pi^2) - (x^2*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (x^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^2*Pi)
- (3*FresnelS[b*x]^2)/(2*b^5*Pi^2) + (3*x*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^4*Pi^2) + Sin[b^2*Pi*x^2]/(b^5
*Pi^3)

Rule 6454

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*b^4)/4] && IGtQ[m, 1]

Rule 6462

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

Rule 3379

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 2634

Int[sin[(c_.) + ((d_.)*(x_))/2]^2, x_Symbol] :> Simp[x/2, x] - Simp[Sin[2*c + d*x]/(2*d), x] /; FreeQ[{c, d},
x]

Rule 6440

Int[FresnelS[(b_.)*(x_)]^(n_.)*Sin[(d_.)*(x_)^2], x_Symbol] :> Dist[(Pi*b)/(2*d), Subst[Int[x^n, x], x, Fresne
lS[b*x]], x] /; FreeQ[{b, d, n}, x] && EqQ[d^2, (Pi^2*b^4)/4]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx &=-\frac{x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{3 \int x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^2 \pi }+\frac{\int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac{x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{3 x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac{3 \int S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^4 \pi ^2}-\frac{3 \int x \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}+\frac{\operatorname{Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi }\\ &=-\frac{x^2 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac{x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{3 x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac{3 \operatorname{Subst}(\int x \, dx,x,S(b x))}{b^5 \pi ^2}+\frac{\operatorname{Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac{3 \operatorname{Subst}\left (\int \sin ^2\left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^3 \pi ^2}\\ &=-\frac{3 x^2}{4 b^3 \pi ^2}-\frac{x^2 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac{x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }-\frac{3 S(b x)^2}{2 b^5 \pi ^2}+\frac{3 x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac{\sin \left (b^2 \pi x^2\right )}{b^5 \pi ^3}\\ \end{align*}

Mathematica [A]  time = 0.0058869, size = 120, normalized size = 1. \[ \frac{3 x S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{x^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{3 S(b x)^2}{2 \pi ^2 b^5}-\frac{3 x^2}{4 \pi ^2 b^3}+\frac{\sin \left (\pi b^2 x^2\right )}{\pi ^3 b^5}-\frac{x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2],x]

[Out]

(-3*x^2)/(4*b^3*Pi^2) - (x^2*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (x^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^2*Pi)
- (3*FresnelS[b*x]^2)/(2*b^5*Pi^2) + (3*x*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^4*Pi^2) + Sin[b^2*Pi*x^2]/(b^5
*Pi^3)

________________________________________________________________________________________

Maple [F]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{x}^{4}{\it FresnelS} \left ( bx \right ) \sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2),x)

[Out]

int(x^4*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4}{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*fresnels(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="maxima")

[Out]

integrate(x^4*fresnels(b*x)*sin(1/2*pi*b^2*x^2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{4}{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*fresnels(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="fricas")

[Out]

integral(x^4*fresnels(b*x)*sin(1/2*pi*b^2*x^2), x)

________________________________________________________________________________________

Sympy [A]  time = 26.5934, size = 151, normalized size = 1.26 \begin{align*} \begin{cases} - \frac{x^{3} \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{\pi b^{2}} - \frac{x^{2} \sin ^{2}{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{2} b^{3}} - \frac{x^{2} \cos ^{2}{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{\pi ^{2} b^{3}} + \frac{3 x \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{\pi ^{2} b^{4}} + \frac{2 \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{\pi ^{3} b^{5}} - \frac{3 S^{2}\left (b x\right )}{2 \pi ^{2} b^{5}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*fresnels(b*x)*sin(1/2*b**2*pi*x**2),x)

[Out]

Piecewise((-x**3*cos(pi*b**2*x**2/2)*fresnels(b*x)/(pi*b**2) - x**2*sin(pi*b**2*x**2/2)**2/(2*pi**2*b**3) - x*
*2*cos(pi*b**2*x**2/2)**2/(pi**2*b**3) + 3*x*sin(pi*b**2*x**2/2)*fresnels(b*x)/(pi**2*b**4) + 2*sin(pi*b**2*x*
*2/2)*cos(pi*b**2*x**2/2)/(pi**3*b**5) - 3*fresnels(b*x)**2/(2*pi**2*b**5), Ne(b, 0)), (0, True))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4}{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*fresnels(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="giac")

[Out]

integrate(x^4*fresnels(b*x)*sin(1/2*pi*b^2*x^2), x)