\(\int x^5 \operatorname {FresnelS}(b x)^2 \, dx\) [33]

Optimal result

Integrand size = 10, antiderivative size = 265 \[ \int x^5 \operatorname {FresnelS}(b x)^2 \, dx=\frac {5 x^4}{24 b^2 \pi ^2}-\frac {11 \cos \left (b^2 \pi x^2\right )}{6 b^6 \pi ^4}+\frac {x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac {5 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^5 \pi ^3}+\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {5 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^6 \pi ^3}+\frac {1}{6} x^6 \operatorname {FresnelS}(b x)^2-\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}-\frac {5 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}-\frac {7 x^2 \sin \left (b^2 \pi x^2\right )}{12 b^4 \pi ^3} \]

[Out]

5/24*x^4/b^2/Pi^2-11/6*cos(b^2*Pi*x^2)/b^6/Pi^4+1/12*x^4*cos(b^2*Pi*x^2)/b^2/Pi^2-5*x*cos(1/2*b^2*Pi*x^2)*Fres
nelS(b*x)/b^5/Pi^3+1/3*x^5*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b/Pi+5/2*FresnelC(b*x)*FresnelS(b*x)/b^6/Pi^3+1/6
*x^6*FresnelS(b*x)^2-5/8*I*x^2*hypergeom([1, 1],[3/2, 2],-1/2*I*b^2*Pi*x^2)/b^4/Pi^3+5/8*I*x^2*hypergeom([1, 1
],[3/2, 2],1/2*I*b^2*Pi*x^2)/b^4/Pi^3-5/3*x^3*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^3/Pi^2-7/12*x^2*sin(b^2*Pi*x
^2)/b^4/Pi^3

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6565, 6589, 6597, 3460, 3390, 30, 3377, 2718, 6581} \[ \int x^5 \operatorname {FresnelS}(b x)^2 \, dx=-\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^3 b^4}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^3 b^4}+\frac {5 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 \pi ^3 b^6}+\frac {x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi b}+\frac {5 x^4}{24 \pi ^2 b^2}+\frac {x^4 \cos \left (\pi b^2 x^2\right )}{12 \pi ^2 b^2}-\frac {11 \cos \left (\pi b^2 x^2\right )}{6 \pi ^4 b^6}-\frac {5 x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^5}-\frac {7 x^2 \sin \left (\pi b^2 x^2\right )}{12 \pi ^3 b^4}-\frac {5 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac {1}{6} x^6 \operatorname {FresnelS}(b x)^2 \]

[In]

Int[x^5*FresnelS[b*x]^2,x]

[Out]

(5*x^4)/(24*b^2*Pi^2) - (11*Cos[b^2*Pi*x^2])/(6*b^6*Pi^4) + (x^4*Cos[b^2*Pi*x^2])/(12*b^2*Pi^2) - (5*x*Cos[(b^
2*Pi*x^2)/2]*FresnelS[b*x])/(b^5*Pi^3) + (x^5*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(3*b*Pi) + (5*FresnelC[b*x]*F
resnelS[b*x])/(2*b^6*Pi^3) + (x^6*FresnelS[b*x]^2)/6 - (((5*I)/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-1/
2*I)*b^2*Pi*x^2])/(b^4*Pi^3) + (((5*I)/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2])/(b^4*Pi^3
) - (5*x^3*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(3*b^3*Pi^2) - (7*x^2*Sin[b^2*Pi*x^2])/(12*b^4*Pi^3)

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 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]

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 {1}{6} x^6 \operatorname {FresnelS}(b x)^2-\frac {1}{3} b \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = \frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {1}{6} x^6 \operatorname {FresnelS}(b x)^2-\frac {\int x^5 \sin \left (b^2 \pi x^2\right ) \, dx}{6 \pi }-\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx}{3 b \pi } \\ & = \frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {1}{6} x^6 \operatorname {FresnelS}(b x)^2-\frac {5 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}+\frac {5 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}+\frac {5 \int x^3 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{3 b^2 \pi ^2}-\frac {\text {Subst}\left (\int x^2 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{12 \pi } \\ & = \frac {x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac {5 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^5 \pi ^3}+\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {1}{6} x^6 \operatorname {FresnelS}(b x)^2-\frac {5 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}+\frac {5 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx}{b^5 \pi ^3}+\frac {5 \int x \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^4 \pi ^3}-\frac {\text {Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{6 b^2 \pi ^2}+\frac {5 \text {Subst}\left (\int x \sin ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{6 b^2 \pi ^2} \\ & = \frac {x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac {5 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^5 \pi ^3}+\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {5 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^6 \pi ^3}+\frac {1}{6} x^6 \operatorname {FresnelS}(b x)^2-\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}-\frac {5 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}-\frac {x^2 \sin \left (b^2 \pi x^2\right )}{6 b^4 \pi ^3}+\frac {\text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{6 b^4 \pi ^3}+\frac {5 \text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^4 \pi ^3}+\frac {5 \text {Subst}\left (\int x \, dx,x,x^2\right )}{12 b^2 \pi ^2}-\frac {5 \text {Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{12 b^2 \pi ^2} \\ & = \frac {5 x^4}{24 b^2 \pi ^2}-\frac {17 \cos \left (b^2 \pi x^2\right )}{12 b^6 \pi ^4}+\frac {x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac {5 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^5 \pi ^3}+\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {5 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^6 \pi ^3}+\frac {1}{6} x^6 \operatorname {FresnelS}(b x)^2-\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}-\frac {5 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}-\frac {7 x^2 \sin \left (b^2 \pi x^2\right )}{12 b^4 \pi ^3}+\frac {5 \text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{12 b^4 \pi ^3} \\ & = \frac {5 x^4}{24 b^2 \pi ^2}-\frac {11 \cos \left (b^2 \pi x^2\right )}{6 b^6 \pi ^4}+\frac {x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac {5 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^5 \pi ^3}+\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {5 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^6 \pi ^3}+\frac {1}{6} x^6 \operatorname {FresnelS}(b x)^2-\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}-\frac {5 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}-\frac {7 x^2 \sin \left (b^2 \pi x^2\right )}{12 b^4 \pi ^3} \\ \end{align*}

Mathematica [F]

\[ \int x^5 \operatorname {FresnelS}(b x)^2 \, dx=\int x^5 \operatorname {FresnelS}(b x)^2 \, dx \]

[In]

Integrate[x^5*FresnelS[b*x]^2,x]

[Out]

Integrate[x^5*FresnelS[b*x]^2, x]

Maple [F]

\[\int x^{5} \operatorname {FresnelS}\left (b x \right )^{2}d x\]

[In]

int(x^5*FresnelS(b*x)^2,x)

[Out]

int(x^5*FresnelS(b*x)^2,x)

Fricas [F]

\[ \int x^5 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{5} \operatorname {S}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^5*fresnel_sin(b*x)^2,x, algorithm="fricas")

[Out]

integral(x^5*fresnel_sin(b*x)^2, x)

Sympy [F]

\[ \int x^5 \operatorname {FresnelS}(b x)^2 \, dx=\int x^{5} S^{2}\left (b x\right )\, dx \]

[In]

integrate(x**5*fresnels(b*x)**2,x)

[Out]

Integral(x**5*fresnels(b*x)**2, x)

Maxima [F]

\[ \int x^5 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{5} \operatorname {S}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^5*fresnel_sin(b*x)^2,x, algorithm="maxima")

[Out]

integrate(x^5*fresnel_sin(b*x)^2, x)

Giac [F]

\[ \int x^5 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{5} \operatorname {S}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^5*fresnel_sin(b*x)^2,x, algorithm="giac")

[Out]

integrate(x^5*fresnel_sin(b*x)^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^5 \operatorname {FresnelS}(b x)^2 \, dx=\int x^5\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \]

[In]

int(x^5*FresnelS(b*x)^2,x)

[Out]

int(x^5*FresnelS(b*x)^2, x)