3.151 \(\int \frac {C(b x)^2}{x^4} \, dx\)

Optimal. Leaf size=120 \[ -\frac {1}{3} \pi b^3 \text {Int}\left (\frac {C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x},x\right )-\frac {\pi b^3 S\left (\sqrt {2} b x\right )}{3 \sqrt {2}}-\frac {b C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^2}-\frac {b^2 \cos \left (\pi b^2 x^2\right )}{6 x}-\frac {b^2}{6 x}-\frac {C(b x)^2}{3 x^3} \]

[Out]

-1/6*b^2/x-1/6*b^2*cos(b^2*Pi*x^2)/x-1/3*b*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2-1/3*FresnelC(b*x)^2/x^3-1/6*b
^3*Pi*FresnelS(b*x*2^(1/2))*2^(1/2)-1/3*b^3*Pi*Unintegrable(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x,x)

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Rubi [A]  time = 0.08, 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 = {} \[ \int \frac {\text {FresnelC}(b x)^2}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[FresnelC[b*x]^2/x^4,x]

[Out]

-b^2/(6*x) - (b^2*Cos[b^2*Pi*x^2])/(6*x) - (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(3*x^2) - FresnelC[b*x]^2/(3*
x^3) - (b^3*Pi*FresnelS[Sqrt[2]*b*x])/(3*Sqrt[2]) - (b^3*Pi*Defer[Int][(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x,
x])/3

Rubi steps

\begin {align*} \int \frac {C(b x)^2}{x^4} \, dx &=-\frac {C(b x)^2}{3 x^3}+\frac {1}{3} (2 b) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^3} \, dx\\ &=-\frac {b^2}{6 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{3 x^2}-\frac {C(b x)^2}{3 x^3}+\frac {1}{6} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{3} \left (b^3 \pi \right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx\\ &=-\frac {b^2}{6 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{6 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{3 x^2}-\frac {C(b x)^2}{3 x^3}-\frac {1}{3} \left (b^3 \pi \right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx-\frac {1}{3} \left (b^4 \pi \right ) \int \sin \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac {b^2}{6 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{6 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{3 x^2}-\frac {C(b x)^2}{3 x^3}-\frac {b^3 \pi S\left (\sqrt {2} b x\right )}{3 \sqrt {2}}-\frac {1}{3} \left (b^3 \pi \right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {C(b x)^2}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[FresnelC[b*x]^2/x^4,x]

[Out]

Integrate[FresnelC[b*x]^2/x^4, x]

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnelc}\left (b x\right )^{2}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)^2/x^4,x, algorithm="fricas")

[Out]

integral(fresnelc(b*x)^2/x^4, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnelc}\left (b x\right )^{2}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)^2/x^4,x, algorithm="giac")

[Out]

integrate(fresnelc(b*x)^2/x^4, x)

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maple [A]  time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {\FresnelC \left (b x \right )^{2}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)^2/x^4,x)

[Out]

int(FresnelC(b*x)^2/x^4,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnelc}\left (b x\right )^{2}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)^2/x^4,x, algorithm="maxima")

[Out]

integrate(fresnelc(b*x)^2/x^4, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)^2/x^4,x)

[Out]

int(FresnelC(b*x)^2/x^4, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {C^{2}\left (b x\right )}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)**2/x**4,x)

[Out]

Integral(fresnelc(b*x)**2/x**4, x)

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