3.2.54 \(\int \frac {\text {FresnelC}(b x)^2}{x^7} \, dx\) [154]

Optimal. Leaf size=166 \[ -\frac {b^2}{120 x^4}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac {1}{72} b^6 \pi ^2 \text {CosIntegral}\left (b^2 \pi x^2\right )-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{15 x^5}-\frac {\text {FresnelC}(b x)^2}{6 x^6}+\frac {b^3 \pi \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{45 x^3}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac {1}{45} b^5 \pi ^2 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^2},x\right ) \]

[Out]

-1/120*b^2/x^4-1/72*b^6*Pi^2*Ci(b^2*Pi*x^2)-1/120*b^2*cos(b^2*Pi*x^2)/x^4-1/15*b*cos(1/2*b^2*Pi*x^2)*FresnelC(
b*x)/x^5-1/6*FresnelC(b*x)^2/x^6+1/45*b^3*Pi*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^3+1/72*b^4*Pi*sin(b^2*Pi*x^2)
/x^2-1/45*b^5*Pi^2*Unintegrable(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2,x)

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Rubi [A]
time = 0.15, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\text {FresnelC}(b x)^2}{x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[FresnelC[b*x]^2/x^7,x]

[Out]

-1/120*b^2/x^4 - (b^2*Cos[b^2*Pi*x^2])/(120*x^4) - (b^6*Pi^2*CosIntegral[b^2*Pi*x^2])/72 - (b*Cos[(b^2*Pi*x^2)
/2]*FresnelC[b*x])/(15*x^5) - FresnelC[b*x]^2/(6*x^6) + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(45*x^3) +
(b^4*Pi*Sin[b^2*Pi*x^2])/(72*x^2) - (b^5*Pi^2*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/45

Rubi steps

\begin {align*} \int \frac {C(b x)^2}{x^7} \, dx &=-\frac {C(b x)^2}{6 x^6}+\frac {1}{3} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^6} \, dx\\ &=-\frac {b^2}{120 x^4}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{15 x^5}-\frac {C(b x)^2}{6 x^6}+\frac {1}{30} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac {1}{15} \left (b^3 \pi \right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=-\frac {b^2}{120 x^4}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{15 x^5}-\frac {C(b x)^2}{6 x^6}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{45 x^3}+\frac {1}{60} b^2 \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac {1}{90} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ &=-\frac {b^2}{120 x^4}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{15 x^5}-\frac {C(b x)^2}{6 x^6}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{45 x^3}-\frac {1}{180} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{120} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ &=-\frac {b^2}{120 x^4}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{15 x^5}-\frac {C(b x)^2}{6 x^6}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{45 x^3}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx-\frac {1}{180} \left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{120} \left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=-\frac {b^2}{120 x^4}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac {1}{72} b^6 \pi ^2 \text {Ci}\left (b^2 \pi x^2\right )-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{15 x^5}-\frac {C(b x)^2}{6 x^6}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{45 x^3}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {FresnelC}(b x)^2}{x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[FresnelC[b*x]^2/x^7,x]

[Out]

Integrate[FresnelC[b*x]^2/x^7, x]

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Maple [A]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\FresnelC \left (b x \right )^{2}}{x^{7}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)^2/x^7,x)

[Out]

int(FresnelC(b*x)^2/x^7,x)

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Maxima [A]
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^7,x, algorithm="maxima")

[Out]

integrate(fresnel_cos(b*x)^2/x^7, x)

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Fricas [A]
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^7,x, algorithm="fricas")

[Out]

integral(fresnel_cos(b*x)^2/x^7, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {C^{2}\left (b x\right )}{x^{7}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)**2/x**7,x)

[Out]

Integral(fresnelc(b*x)**2/x**7, x)

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Giac [A]
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^7,x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)^2/x^7, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)^2/x^7,x)

[Out]

int(FresnelC(b*x)^2/x^7, x)

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