\(\int x^4 \cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelS}(b x) \, dx\) [95]

Optimal result

Integrand size = 20, antiderivative size = 195 \[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=-\frac {x^4}{8 b \pi }+\frac {\cos \left (b^2 \pi x^2\right )}{b^5 \pi ^3}+\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}-\frac {3 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^5 \pi ^2}+\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}-\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x^2 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \]

[Out]

-1/8*x^4/b/Pi+cos(b^2*Pi*x^2)/b^5/Pi^3+3*x*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b^4/Pi^2-3/2*FresnelC(b*x)*Fresne
lS(b*x)/b^5/Pi^2+3/8*I*x^2*hypergeom([1, 1],[3/2, 2],-1/2*I*b^2*Pi*x^2)/b^3/Pi^2-3/8*I*x^2*hypergeom([1, 1],[3
/2, 2],1/2*I*b^2*Pi*x^2)/b^3/Pi^2+x^3*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^2/Pi+1/4*x^2*sin(b^2*Pi*x^2)/b^3/Pi^
2

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6597, 3460, 3390, 30, 3377, 2718, 6589, 6581} \[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^2 b^3}-\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^2 b^3}-\frac {3 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 \pi ^2 b^5}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^3 b^5}+\frac {3 x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^2 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac {x^4}{8 \pi b} \]

[In]

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

[Out]

-1/8*x^4/(b*Pi) + Cos[b^2*Pi*x^2]/(b^5*Pi^3) + (3*x*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^4*Pi^2) - (3*Fresnel
C[b*x]*FresnelS[b*x])/(2*b^5*Pi^2) + (((3*I)/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-1/2*I)*b^2*Pi*x^2])/
(b^3*Pi^2) - (((3*I)/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2])/(b^3*Pi^2) + (x^3*FresnelS[
b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) + (x^2*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)

Rule 30

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

Rule 2718

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

Rule 3377

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 3390

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 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 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]

Rule 6597

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^2/4)*b^4] && IGtQ[m, 1]

Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^3 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi } \\ & = \frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {3 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx}{b^4 \pi ^2}-\frac {3 \int x \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^3 \pi ^2}-\frac {\text {Subst}\left (\int x \sin ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b \pi } \\ & = \frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}-\frac {3 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^5 \pi ^2}+\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}-\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {3 \text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac {\text {Subst}\left (\int x \, dx,x,x^2\right )}{4 b \pi }+\frac {\text {Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi } \\ & = -\frac {x^4}{8 b \pi }+\frac {3 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}-\frac {3 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^5 \pi ^2}+\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}-\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x^2 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {\text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2} \\ & = -\frac {x^4}{8 b \pi }+\frac {\cos \left (b^2 \pi x^2\right )}{b^5 \pi ^3}+\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}-\frac {3 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^5 \pi ^2}+\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}-\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x^2 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \\ \end{align*}

Mathematica [F]

\[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx \]

[In]

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

[Out]

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

Maple [F]

\[\int x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelS}\left (b x \right )d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{4} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \]

[In]

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

[Out]

integral(x^4*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)

Sympy [F]

\[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^{4} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \]

[In]

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

[Out]

Integral(x**4*cos(pi*b**2*x**2/2)*fresnels(b*x), x)

Maxima [F]

\[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{4} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \]

[In]

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

[Out]

integrate(x^4*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)

Giac [F]

\[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{4} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \]

[In]

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

[Out]

integrate(x^4*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)

Mupad [F(-1)]

Timed out. \[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^4\,\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]

[In]

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

[Out]

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