3.45 \(\int \frac {S(b x)^2}{x^7} \, dx\)

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

[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^4-1/45*b^3*Pi*cos(1/2*b^2*Pi*x^2)*Fres
nelS(b*x)/x^3-1/6*FresnelS(b*x)^2/x^6-1/15*b*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^5-1/72*b^4*Pi*sin(b^2*Pi*x^2)
/x^2-1/45*b^5*Pi^2*Unintegrable(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^2,x)

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Rubi [A]  time = 0.22, 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 = {} \[ \int \frac {S(b x)^2}{x^7} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {S(b x)^2}{x^7} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[FresnelS[b*x]^2/x^7, x]

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fricas [A]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right )^{2}}{x^{7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^7,x, algorithm="fricas")

[Out]

integral(fresnels(b*x)^2/x^7, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnels}\left (b x\right )^{2}}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^7,x, algorithm="giac")

[Out]

integrate(fresnels(b*x)^2/x^7, x)

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maple [A]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {S}\left (b x \right )^{2}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnels}\left (b x\right )^{2}}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^7,x, algorithm="maxima")

[Out]

integrate(fresnels(b*x)^2/x^7, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {FresnelS}\left (b\,x\right )}^2}{x^7} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(FresnelS(b*x)^2/x^7, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(fresnels(b*x)**2/x**7, x)

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