\(\int \frac {\operatorname {FresnelS}(b x) \sin (\frac {1}{2} b^2 \pi x^2)}{x^8} \, dx\) [87]

Optimal result

Integrand size = 20, antiderivative size = 224 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=-\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{105 x}+\frac {1}{210} b^7 \pi ^4 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{70} b^7 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \]

[Out]

-1/84*b/x^6+1/420*b^5*Pi^2/x^2+1/84*b*cos(b^2*Pi*x^2)/x^6-1/84*b^5*Pi^2*cos(b^2*Pi*x^2)/x^2-1/35*b^2*Pi*cos(1/
2*b^2*Pi*x^2)*FresnelS(b*x)/x^5+1/105*b^6*Pi^3*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x+1/210*b^7*Pi^4*FresnelS(b*x
)^2-1/70*b^7*Pi^3*Si(b^2*Pi*x^2)-1/7*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^7+1/105*b^4*Pi^2*FresnelS(b*x)*sin(1/
2*b^2*Pi*x^2)/x^3-1/105*b^3*Pi*sin(b^2*Pi*x^2)/x^4

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6591, 6599, 6575, 30, 3456, 3461, 3378, 3380, 3460} \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\frac {1}{210} \pi ^4 b^7 \operatorname {FresnelS}(b x)^2+\frac {\pi ^2 b^5}{420 x^2}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {\pi b^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{35 x^5}+\frac {b \cos \left (\pi b^2 x^2\right )}{84 x^6}-\frac {1}{70} \pi ^3 b^7 \text {Si}\left (b^2 \pi x^2\right )+\frac {\pi ^3 b^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{105 x}-\frac {\pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{84 x^2}+\frac {\pi ^2 b^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac {\pi b^3 \sin \left (\pi b^2 x^2\right )}{105 x^4}-\frac {b}{84 x^6} \]

[In]

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

[Out]

-1/84*b/x^6 + (b^5*Pi^2)/(420*x^2) + (b*Cos[b^2*Pi*x^2])/(84*x^6) - (b^5*Pi^2*Cos[b^2*Pi*x^2])/(84*x^2) - (b^2
*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(35*x^5) + (b^6*Pi^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(105*x) + (b^7*
Pi^4*FresnelS[b*x]^2)/210 - (FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(7*x^7) + (b^4*Pi^2*FresnelS[b*x]*Sin[(b^2*Pi*
x^2)/2])/(105*x^3) - (b^3*Pi*Sin[b^2*Pi*x^2])/(105*x^4) - (b^7*Pi^3*SinIntegral[b^2*Pi*x^2])/70

Rule 30

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

Rule 3378

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

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3456

Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, 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 3461

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[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 6575

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^2/4)*b^4]

Rule 6591

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

Rule 6599

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

Rubi steps \begin{align*} \text {integral}& = -\frac {b}{84 x^6}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}-\frac {1}{14} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^7} \, dx+\frac {1}{7} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx \\ & = -\frac {b}{84 x^6}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}-\frac {1}{28} b \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac {1}{70} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac {1}{35} \left (b^4 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \\ & = -\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}+\frac {1}{140} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{84} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{210} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{105} \left (b^6 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx \\ & = -\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{105 x}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}+\frac {1}{420} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{280} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{168} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{210} \left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx+\frac {1}{105} \left (b^8 \pi ^4\right ) \int \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = -\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{105 x}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{420} b^7 \pi ^3 \text {Si}\left (b^2 \pi x^2\right )-\frac {1}{420} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{280} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{168} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac {1}{105} \left (b^7 \pi ^4\right ) \text {Subst}(\int x \, dx,x,\operatorname {FresnelS}(b x)) \\ & = -\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{105 x}+\frac {1}{210} b^7 \pi ^4 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{70} b^7 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=-\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{105 x}+\frac {1}{210} b^7 \pi ^4 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{70} b^7 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \]

[In]

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

[Out]

-1/84*b/x^6 + (b^5*Pi^2)/(420*x^2) + (b*Cos[b^2*Pi*x^2])/(84*x^6) - (b^5*Pi^2*Cos[b^2*Pi*x^2])/(84*x^2) - (b^2
*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(35*x^5) + (b^6*Pi^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(105*x) + (b^7*
Pi^4*FresnelS[b*x]^2)/210 - (FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(7*x^7) + (b^4*Pi^2*FresnelS[b*x]*Sin[(b^2*Pi*
x^2)/2])/(105*x^3) - (b^3*Pi*Sin[b^2*Pi*x^2])/(105*x^4) - (b^7*Pi^3*SinIntegral[b^2*Pi*x^2])/70

Maple [F]

\[\int \frac {\operatorname {FresnelS}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{8}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.77 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\frac {\pi ^{4} b^{7} x^{7} \operatorname {S}\left (b x\right )^{2} - 3 \, \pi ^{3} b^{7} x^{7} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) + 3 \, \pi ^{2} b^{5} x^{5} - 5 \, {\left (\pi ^{2} b^{5} x^{5} - b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 2 \, {\left (\pi ^{3} b^{6} x^{6} - 3 \, \pi b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - 5 \, b x - 2 \, {\left (2 \, \pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{4} x^{4} - 15\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{210 \, x^{7}} \]

[In]

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

[Out]

1/210*(pi^4*b^7*x^7*fresnel_sin(b*x)^2 - 3*pi^3*b^7*x^7*sin_integral(pi*b^2*x^2) + 3*pi^2*b^5*x^5 - 5*(pi^2*b^
5*x^5 - b*x)*cos(1/2*pi*b^2*x^2)^2 + 2*(pi^3*b^6*x^6 - 3*pi*b^2*x^2)*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x) - 5*
b*x - 2*(2*pi*b^3*x^3*cos(1/2*pi*b^2*x^2) - (pi^2*b^4*x^4 - 15)*fresnel_sin(b*x))*sin(1/2*pi*b^2*x^2))/x^7

Sympy [F]

\[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{8}}\, dx \]

[In]

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

[Out]

Integral(sin(pi*b**2*x**2/2)*fresnels(b*x)/x**8, x)

Maxima [F]

\[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}} \,d x } \]

[In]

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

[Out]

integrate(fresnel_sin(b*x)*sin(1/2*pi*b^2*x^2)/x^8, x)

Giac [F]

\[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}} \,d x } \]

[In]

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

[Out]

integrate(fresnel_sin(b*x)*sin(1/2*pi*b^2*x^2)/x^8, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^8} \,d x \]

[In]

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

[Out]

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