3.2.56 \(\int \frac {\text {FresnelC}(b x)^2}{x^9} \, dx\) [156]

Optimal. Leaf size=242 \[ -\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}+\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{28 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{420 x^3}+\frac {1}{840} b^8 \pi ^4 \text {FresnelC}(b x)^2-\frac {\text {FresnelC}(b x)^2}{8 x^8}+\frac {b^3 \pi \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac {b^7 \pi ^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}+\frac {1}{280} b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \]

[Out]

-1/336*b^2/x^6+1/1680*b^6*Pi^2/x^2-1/336*b^2*cos(b^2*Pi*x^2)/x^6+1/336*b^6*Pi^2*cos(b^2*Pi*x^2)/x^2-1/28*b*cos
(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^7+1/420*b^5*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^3+1/840*b^8*Pi^4*Fresnel
C(b*x)^2-1/8*FresnelC(b*x)^2/x^8+1/280*b^8*Pi^3*Si(b^2*Pi*x^2)+1/140*b^3*Pi*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/
x^5-1/420*b^7*Pi^3*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x+1/420*b^4*Pi*sin(b^2*Pi*x^2)/x^4

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Rubi [A]
time = 0.27, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6566, 6592, 6600, 6576, 30, 3456, 3461, 3378, 3380, 3460} \begin {gather*} \frac {1}{840} \pi ^4 b^8 \text {FresnelC}(b x)^2+\frac {\pi ^2 b^6}{1680 x^2}-\frac {b \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{28 x^7}-\frac {b^2}{336 x^6}-\frac {b^2 \cos \left (\pi b^2 x^2\right )}{336 x^6}+\frac {1}{280} \pi ^3 b^8 \text {Si}\left (b^2 \pi x^2\right )-\frac {\pi ^3 b^7 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{420 x}+\frac {\pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{336 x^2}+\frac {\pi ^2 b^5 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{420 x^3}+\frac {\pi b^4 \sin \left (\pi b^2 x^2\right )}{420 x^4}+\frac {\pi b^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{140 x^5}-\frac {\text {FresnelC}(b x)^2}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[FresnelC[b*x]^2/x^9,x]

[Out]

-1/336*b^2/x^6 + (b^6*Pi^2)/(1680*x^2) - (b^2*Cos[b^2*Pi*x^2])/(336*x^6) + (b^6*Pi^2*Cos[b^2*Pi*x^2])/(336*x^2
) - (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(28*x^7) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(420*x^3) +
(b^8*Pi^4*FresnelC[b*x]^2)/840 - FresnelC[b*x]^2/(8*x^8) + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(140*x^5
) - (b^7*Pi^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(420*x) + (b^4*Pi*Sin[b^2*Pi*x^2])/(420*x^4) + (b^8*Pi^3*SinI
ntegral[b^2*Pi*x^2])/280

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 6566

Int[FresnelC[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(FresnelC[b*x]^2/(m + 1)), x] - Dist[2*(b/(
m + 1)), Int[x^(m + 1)*Cos[(Pi/2)*b^2*x^2]*FresnelC[b*x], x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]

Rule 6576

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

Rule 6592

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

Rule 6600

Int[FresnelC[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[x^(m + 1)*Sin[d*x^2]*(FresnelC[b*x]/(m
+ 1)), x] + (-Dist[2*(d/(m + 1)), Int[x^(m + 2)*Cos[d*x^2]*FresnelC[b*x], x], x] - Dist[b/(2*(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*} \int \frac {C(b x)^2}{x^9} \, dx &=-\frac {C(b x)^2}{8 x^8}+\frac {1}{4} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^8} \, dx\\ &=-\frac {b^2}{336 x^6}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}-\frac {C(b x)^2}{8 x^8}+\frac {1}{56} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^7} \, dx-\frac {1}{28} \left (b^3 \pi \right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6} \, dx\\ &=-\frac {b^2}{336 x^6}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}-\frac {C(b x)^2}{8 x^8}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}+\frac {1}{112} b^2 \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )-\frac {1}{280} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac {1}{140} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^4} \, dx\\ &=-\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{420 x^3}-\frac {C(b x)^2}{8 x^8}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac {1}{560} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac {1}{336} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac {1}{840} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx+\frac {1}{420} \left (b^7 \pi ^3\right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{420 x^3}-\frac {C(b x)^2}{8 x^8}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac {b^7 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}-\frac {\left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1680}-\frac {\left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1120}-\frac {1}{672} \left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{840} \left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx+\frac {1}{420} \left (b^9 \pi ^4\right ) \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=-\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}+\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{420 x^3}-\frac {C(b x)^2}{8 x^8}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac {b^7 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}+\frac {b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right )}{1680}+\frac {\left (b^8 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1680}+\frac {\left (b^8 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1120}+\frac {1}{672} \left (b^8 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac {1}{420} \left (b^8 \pi ^4\right ) \text {Subst}(\int x \, dx,x,C(b x))\\ &=-\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}+\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{420 x^3}+\frac {1}{840} b^8 \pi ^4 C(b x)^2-\frac {C(b x)^2}{8 x^8}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac {b^7 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}+\frac {1}{280} b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 242, normalized size = 1.00 \begin {gather*} -\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}+\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{28 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{420 x^3}+\frac {1}{840} b^8 \pi ^4 \text {FresnelC}(b x)^2-\frac {\text {FresnelC}(b x)^2}{8 x^8}+\frac {b^3 \pi \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac {b^7 \pi ^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}+\frac {1}{280} b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[FresnelC[b*x]^2/x^9,x]

[Out]

-1/336*b^2/x^6 + (b^6*Pi^2)/(1680*x^2) - (b^2*Cos[b^2*Pi*x^2])/(336*x^6) + (b^6*Pi^2*Cos[b^2*Pi*x^2])/(336*x^2
) - (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(28*x^7) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(420*x^3) +
(b^8*Pi^4*FresnelC[b*x]^2)/840 - FresnelC[b*x]^2/(8*x^8) + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(140*x^5
) - (b^7*Pi^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(420*x) + (b^4*Pi*Sin[b^2*Pi*x^2])/(420*x^4) + (b^8*Pi^3*SinI
ntegral[b^2*Pi*x^2])/280

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\FresnelC \left (b x \right )^{2}}{x^{9}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)^2/x^9,x)

[Out]

int(FresnelC(b*x)^2/x^9,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_cos(b*x)^2/x^9,x, algorithm="maxima")

[Out]

integrate(fresnel_cos(b*x)^2/x^9, x)

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Fricas [A]
time = 0.38, size = 178, normalized size = 0.74 \begin {gather*} \frac {3 \, \pi ^{3} b^{8} x^{8} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 2 \, \pi ^{2} b^{6} x^{6} + 5 \, {\left (\pi ^{2} b^{6} x^{6} - b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 2 \, {\left (\pi ^{2} b^{5} x^{5} - 15 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + {\left (\pi ^{4} b^{8} x^{8} - 105\right )} \operatorname {C}\left (b x\right )^{2} + 2 \, {\left (2 \, \pi b^{4} x^{4} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{3} b^{7} x^{7} - 3 \, \pi b^{3} x^{3}\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{840 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_cos(b*x)^2/x^9,x, algorithm="fricas")

[Out]

1/840*(3*pi^3*b^8*x^8*sin_integral(pi*b^2*x^2) - 2*pi^2*b^6*x^6 + 5*(pi^2*b^6*x^6 - b^2*x^2)*cos(1/2*pi*b^2*x^
2)^2 + 2*(pi^2*b^5*x^5 - 15*b*x)*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x) + (pi^4*b^8*x^8 - 105)*fresnel_cos(b*x)^
2 + 2*(2*pi*b^4*x^4*cos(1/2*pi*b^2*x^2) - (pi^3*b^7*x^7 - 3*pi*b^3*x^3)*fresnel_cos(b*x))*sin(1/2*pi*b^2*x^2))
/x^8

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {C^{2}\left (b x\right )}{x^{9}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)**2/x**9,x)

[Out]

Integral(fresnelc(b*x)**2/x**9, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_cos(b*x)^2/x^9,x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)^2/x^9, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)^2/x^9,x)

[Out]

int(FresnelC(b*x)^2/x^9, x)

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