3.85 \(\int \frac{S(b x) \sin (\frac{1}{2} b^2 \pi x^2)}{x^6} \, dx\)

Optimal. Leaf size=147 \[ -\frac{1}{15} \pi ^2 b^4 \text{Unintegrable}\left (\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x^2},x\right )+\frac{1}{24} \pi ^2 b^5 \text{CosIntegral}\left (\pi b^2 x^2\right )-\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac{\pi b^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{15 x^3}-\frac{\pi b^3 \sin \left (\pi b^2 x^2\right )}{24 x^2}+\frac{b \cos \left (\pi b^2 x^2\right )}{40 x^4}-\frac{b}{40 x^4} \]

[Out]

-b/(40*x^4) + (b*Cos[b^2*Pi*x^2])/(40*x^4) + (b^5*Pi^2*CosIntegral[b^2*Pi*x^2])/24 - (b^2*Pi*Cos[(b^2*Pi*x^2)/
2]*FresnelS[b*x])/(15*x^3) - (FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(5*x^5) - (b^3*Pi*Sin[b^2*Pi*x^2])/(24*x^2) -
(b^4*Pi^2*Unintegrable[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x])/15

________________________________________________________________________________________

Rubi [A]  time = 0.192541, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^6} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^6,x]

[Out]

-b/(40*x^4) + (b*Cos[b^2*Pi*x^2])/(40*x^4) + (b^5*Pi^2*CosIntegral[b^2*Pi*x^2])/24 - (b^2*Pi*Cos[(b^2*Pi*x^2)/
2]*FresnelS[b*x])/(15*x^3) - (FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(5*x^5) - (b^3*Pi*Sin[b^2*Pi*x^2])/(24*x^2) -
(b^4*Pi^2*Defer[Int][(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x])/15

Rubi steps

\begin{align*} \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^6} \, dx &=-\frac{b}{40 x^4}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 x^5}-\frac{1}{10} b \int \frac{\cos \left (b^2 \pi x^2\right )}{x^5} \, dx+\frac{1}{5} \left (b^2 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^4} \, dx\\ &=-\frac{b}{40 x^4}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{15 x^3}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 x^5}-\frac{1}{20} b \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac{1}{30} \left (b^3 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac{1}{15} \left (b^4 \pi ^2\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac{b}{40 x^4}+\frac{b \cos \left (b^2 \pi x^2\right )}{40 x^4}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{15 x^3}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 x^5}+\frac{1}{60} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac{1}{40} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{15} \left (b^4 \pi ^2\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac{b}{40 x^4}+\frac{b \cos \left (b^2 \pi x^2\right )}{40 x^4}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{15 x^3}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 x^5}-\frac{b^3 \pi \sin \left (b^2 \pi x^2\right )}{24 x^2}-\frac{1}{15} \left (b^4 \pi ^2\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx+\frac{1}{60} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac{1}{40} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=-\frac{b}{40 x^4}+\frac{b \cos \left (b^2 \pi x^2\right )}{40 x^4}+\frac{1}{24} b^5 \pi ^2 \text{Ci}\left (b^2 \pi x^2\right )-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{15 x^3}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 x^5}-\frac{b^3 \pi \sin \left (b^2 \pi x^2\right )}{24 x^2}-\frac{1}{15} \left (b^4 \pi ^2\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0314466, size = 0, normalized size = 0. \[ \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^6} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^6,x]

[Out]

Integrate[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^6, x]

________________________________________________________________________________________

Maple [A]  time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{6}}\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^6,x)

[Out]

int(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^6,x)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)*sin(1/2*b^2*pi*x^2)/x^6,x, algorithm="maxima")

[Out]

integrate(fresnels(b*x)*sin(1/2*pi*b^2*x^2)/x^6, x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)*sin(1/2*b^2*pi*x^2)/x^6,x, algorithm="fricas")

[Out]

integral(fresnels(b*x)*sin(1/2*pi*b^2*x^2)/x^6, x)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)*sin(1/2*b**2*pi*x**2)/x**6,x)

[Out]

Integral(sin(pi*b**2*x**2/2)*fresnels(b*x)/x**6, x)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)*sin(1/2*b^2*pi*x^2)/x^6,x, algorithm="giac")

[Out]

integrate(fresnels(b*x)*sin(1/2*pi*b^2*x^2)/x^6, x)