3.91 \(\int x^8 \cos (\frac{1}{2} b^2 \pi x^2) S(b x) \, dx\)

Optimal. Leaf size=307 \[ -\frac{105 i x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-\frac{1}{2} i \pi b^2 x^2\right )}{8 \pi ^4 b^7}+\frac{105 i x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},\frac{1}{2} i \pi b^2 x^2\right )}{8 \pi ^4 b^7}+\frac{105 \text{FresnelC}(b x) S(b x)}{2 \pi ^4 b^9}+\frac{x^7 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{35 x^3 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac{7 x^5 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{105 x S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}+\frac{35 x^4}{8 \pi ^3 b^5}+\frac{x^6 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{55 x^2 \sin \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7}+\frac{5 x^4 \cos \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}-\frac{40 \cos \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac{x^8}{16 \pi b} \]

[Out]

(35*x^4)/(8*b^5*Pi^3) - x^8/(16*b*Pi) - (40*Cos[b^2*Pi*x^2])/(b^9*Pi^5) + (5*x^4*Cos[b^2*Pi*x^2])/(2*b^5*Pi^3)
- (105*x*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^8*Pi^4) + (7*x^5*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^4*Pi^2)
+ (105*FresnelC[b*x]*FresnelS[b*x])/(2*b^9*Pi^4) - (((105*I)/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-I/2
)*b^2*Pi*x^2])/(b^7*Pi^4) + (((105*I)/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2])/(b^7*Pi^4)
- (35*x^3*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^6*Pi^3) + (x^7*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) -
(55*x^2*Sin[b^2*Pi*x^2])/(4*b^7*Pi^4) + (x^6*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)

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Rubi [A]  time = 0.441621, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6462, 3379, 3309, 30, 3296, 2638, 6454, 6446} \[ -\frac{105 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i b^2 \pi x^2\right )}{8 \pi ^4 b^7}+\frac{105 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i b^2 \pi x^2\right )}{8 \pi ^4 b^7}+\frac{105 \text{FresnelC}(b x) S(b x)}{2 \pi ^4 b^9}+\frac{x^7 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{35 x^3 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac{7 x^5 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{105 x S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}+\frac{35 x^4}{8 \pi ^3 b^5}+\frac{x^6 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{55 x^2 \sin \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7}+\frac{5 x^4 \cos \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}-\frac{40 \cos \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac{x^8}{16 \pi b} \]

Antiderivative was successfully verified.

[In]

Int[x^8*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x],x]

[Out]

(35*x^4)/(8*b^5*Pi^3) - x^8/(16*b*Pi) - (40*Cos[b^2*Pi*x^2])/(b^9*Pi^5) + (5*x^4*Cos[b^2*Pi*x^2])/(2*b^5*Pi^3)
- (105*x*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^8*Pi^4) + (7*x^5*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^4*Pi^2)
+ (105*FresnelC[b*x]*FresnelS[b*x])/(2*b^9*Pi^4) - (((105*I)/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-I/2
)*b^2*Pi*x^2])/(b^7*Pi^4) + (((105*I)/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2])/(b^7*Pi^4)
- (35*x^3*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^6*Pi^3) + (x^7*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) -
(55*x^2*Sin[b^2*Pi*x^2])/(4*b^7*Pi^4) + (x^6*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)

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 3309

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + ((f_.)*(x_))/2]^2, x_Symbol] :> Dist[1/2, Int[(c + d*x)^m, x], x] -
Dist[1/2, Int[(c + d*x)^m*Cos[2*e + f*x], x], x] /; FreeQ[{c, d, e, f, m}, x]

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 2638

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

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 6446

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

Rubi steps

\begin{align*} \int x^8 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx &=\frac{x^7 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{7 \int x^6 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac{\int x^7 \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=\frac{7 x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac{x^7 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{35 \int x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^4 \pi ^2}-\frac{7 \int x^5 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^3 \pi ^2}-\frac{\operatorname{Subst}\left (\int x^3 \sin ^2\left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b \pi }\\ &=\frac{7 x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac{35 x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^7 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac{105 \int x^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^6 \pi ^3}+\frac{35 \int x^3 \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac{7 \operatorname{Subst}\left (\int x^2 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,x^2\right )}{4 b \pi }+\frac{\operatorname{Subst}\left (\int x^3 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi }\\ &=-\frac{x^8}{16 b \pi }+\frac{7 x^4 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}-\frac{105 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^8 \pi ^4}+\frac{7 x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac{35 x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^7 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac{x^6 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{105 \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^8 \pi ^4}+\frac{105 \int x \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^7 \pi ^4}-\frac{7 \operatorname{Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}+\frac{35 \operatorname{Subst}\left (\int x \sin ^2\left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}-\frac{3 \operatorname{Subst}\left (\int x^2 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}\\ &=-\frac{x^8}{16 b \pi }+\frac{5 x^4 \cos \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac{105 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^8 \pi ^4}+\frac{7 x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac{105 C(b x) S(b x)}{2 b^9 \pi ^4}-\frac{105 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}+\frac{105 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac{35 x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^7 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{7 x^2 \sin \left (b^2 \pi x^2\right )}{2 b^7 \pi ^4}+\frac{x^6 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{7 \operatorname{Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}+\frac{105 \operatorname{Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^7 \pi ^4}-\frac{3 \operatorname{Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}+\frac{35 \operatorname{Subst}\left (\int x \, dx,x,x^2\right )}{4 b^5 \pi ^3}-\frac{35 \operatorname{Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^5 \pi ^3}\\ &=\frac{35 x^4}{8 b^5 \pi ^3}-\frac{x^8}{16 b \pi }-\frac{119 \cos \left (b^2 \pi x^2\right )}{4 b^9 \pi ^5}+\frac{5 x^4 \cos \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac{105 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^8 \pi ^4}+\frac{7 x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac{105 C(b x) S(b x)}{2 b^9 \pi ^4}-\frac{105 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}+\frac{105 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac{35 x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^7 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{55 x^2 \sin \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac{x^6 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{3 \operatorname{Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}+\frac{35 \operatorname{Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^7 \pi ^4}\\ &=\frac{35 x^4}{8 b^5 \pi ^3}-\frac{x^8}{16 b \pi }-\frac{40 \cos \left (b^2 \pi x^2\right )}{b^9 \pi ^5}+\frac{5 x^4 \cos \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac{105 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^8 \pi ^4}+\frac{7 x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac{105 C(b x) S(b x)}{2 b^9 \pi ^4}-\frac{105 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}+\frac{105 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac{35 x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^7 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{55 x^2 \sin \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac{x^6 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end{align*}

Mathematica [F]  time = 0.0459479, size = 0, normalized size = 0. \[ \int x^8 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^8*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x],x]

[Out]

Integrate[x^8*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x], x]

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Maple [F]  time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{x}^{8}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ){\it FresnelS} \left ( bx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)

[Out]

int(x^8*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{8} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*cos(1/2*b^2*pi*x^2)*fresnels(b*x),x, algorithm="maxima")

[Out]

integrate(x^8*cos(1/2*pi*b^2*x^2)*fresnels(b*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{8} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*cos(1/2*b^2*pi*x^2)*fresnels(b*x),x, algorithm="fricas")

[Out]

integral(x^8*cos(1/2*pi*b^2*x^2)*fresnels(b*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*cos(1/2*b**2*pi*x**2)*fresnels(b*x),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{8} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*cos(1/2*b^2*pi*x^2)*fresnels(b*x),x, algorithm="giac")

[Out]

integrate(x^8*cos(1/2*pi*b^2*x^2)*fresnels(b*x), x)