3.3.12 \(\int \frac {\text {FresnelC}(b x) \sin (\frac {1}{2} b^2 \pi x^2)}{x^4} \, dx\) [212]

Optimal. Leaf size=89 \[ \frac {1}{12} b^3 \pi \text {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{12 x^2}+\frac {1}{3} b^2 \pi \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^2},x\right ) \]

[Out]

1/12*b^3*Pi*Ci(b^2*Pi*x^2)-1/3*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^3-1/12*b*sin(b^2*Pi*x^2)/x^2+1/3*b^2*Pi*Uni
ntegrable(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2,x)

________________________________________________________________________________________

Rubi [A]
time = 0.06, 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) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^4,x]

[Out]

(b^3*Pi*CosIntegral[b^2*Pi*x^2])/12 - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(3*x^3) - (b*Sin[b^2*Pi*x^2])/(12*x^
2) + (b^2*Pi*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/3

Rubi steps

\begin {align*} \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx &=-\frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}+\frac {1}{6} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3} \, dx+\frac {1}{3} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ &=-\frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}+\frac {1}{12} b \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{3} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ &=-\frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{12 x^2}+\frac {1}{3} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx+\frac {1}{12} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=\frac {1}{12} b^3 \pi \text {Ci}\left (b^2 \pi x^2\right )-\frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{12 x^2}+\frac {1}{3} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^4,x]

[Out]

Integrate[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^4, x]

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^4,x)

[Out]

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^4,x)

________________________________________________________________________________________

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)*sin(1/2*b^2*pi*x^2)/x^4,x, algorithm="maxima")

[Out]

integrate(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2)/x^4, x)

________________________________________________________________________________________

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)*sin(1/2*b^2*pi*x^2)/x^4,x, algorithm="fricas")

[Out]

integral(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2)/x^4, x)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)*sin(1/2*b**2*pi*x**2)/x**4,x)

[Out]

Integral(sin(pi*b**2*x**2/2)*fresnelc(b*x)/x**4, x)

________________________________________________________________________________________

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)*sin(1/2*b^2*pi*x^2)/x^4,x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2)/x^4, x)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________