\(\int \frac {\operatorname {FresnelS}(b x)^2}{x^6} \, dx\) [44]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^6} \, dx=-\frac {b^2}{60 x^3}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{60 x^3}+\frac {7 b^5 \pi ^2 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{60 \sqrt {2}}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{20 x^2}-\frac {\operatorname {FresnelS}(b x)^2}{5 x^5}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{10 x^4}-\frac {7 b^4 \pi \sin \left (b^2 \pi x^2\right )}{120 x}-\frac {1}{20} b^5 \pi ^2 \text {Int}\left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \]

[Out]

-1/60*b^2/x^3+1/60*b^2*cos(b^2*Pi*x^2)/x^3-1/20*b^3*Pi*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^2-1/5*FresnelS(b*x)
^2/x^5-1/10*b*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^4-7/120*b^4*Pi*sin(b^2*Pi*x^2)/x+7/120*b^5*Pi^2*FresnelC(b*x
*2^(1/2))*2^(1/2)-1/20*b^5*Pi^2*Unintegrable(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x,x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^6} \, dx=\int \frac {\operatorname {FresnelS}(b x)^2}{x^6} \, dx \]

[In]

Int[FresnelS[b*x]^2/x^6,x]

[Out]

-1/60*b^2/x^3 + (b^2*Cos[b^2*Pi*x^2])/(60*x^3) + (7*b^5*Pi^2*FresnelC[Sqrt[2]*b*x])/(60*Sqrt[2]) - (b^3*Pi*Cos
[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(20*x^2) - FresnelS[b*x]^2/(5*x^5) - (b*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(10
*x^4) - (7*b^4*Pi*Sin[b^2*Pi*x^2])/(120*x) - (b^5*Pi^2*Defer[Int][(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/2
0

Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelS}(b x)^2}{5 x^5}+\frac {1}{5} (2 b) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx \\ & = -\frac {b^2}{60 x^3}-\frac {\operatorname {FresnelS}(b x)^2}{5 x^5}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{10 x^4}-\frac {1}{20} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac {1}{10} \left (b^3 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3} \, dx \\ & = -\frac {b^2}{60 x^3}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{60 x^3}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{20 x^2}-\frac {\operatorname {FresnelS}(b x)^2}{5 x^5}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{10 x^4}+\frac {1}{40} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{30} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{20} \left (b^5 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ & = -\frac {b^2}{60 x^3}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{60 x^3}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{20 x^2}-\frac {\operatorname {FresnelS}(b x)^2}{5 x^5}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{10 x^4}-\frac {7 b^4 \pi \sin \left (b^2 \pi x^2\right )}{120 x}-\frac {1}{20} \left (b^5 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx+\frac {1}{20} \left (b^6 \pi ^2\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx+\frac {1}{15} \left (b^6 \pi ^2\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx \\ & = -\frac {b^2}{60 x^3}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{60 x^3}+\frac {7 b^5 \pi ^2 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{60 \sqrt {2}}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{20 x^2}-\frac {\operatorname {FresnelS}(b x)^2}{5 x^5}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{10 x^4}-\frac {7 b^4 \pi \sin \left (b^2 \pi x^2\right )}{120 x}-\frac {1}{20} \left (b^5 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^6} \, dx=\int \frac {\operatorname {FresnelS}(b x)^2}{x^6} \, dx \]

[In]

Integrate[FresnelS[b*x]^2/x^6,x]

[Out]

Integrate[FresnelS[b*x]^2/x^6, x]

Maple [N/A] (verified)

Not integrable

Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {FresnelS}\left (b x \right )^{2}}{x^{6}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^6} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{6}} \,d x } \]

[In]

integrate(fresnel_sin(b*x)^2/x^6,x, algorithm="fricas")

[Out]

integral(fresnel_sin(b*x)^2/x^6, x)

Sympy [N/A]

Not integrable

Time = 1.40 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^6} \, dx=\int \frac {S^{2}\left (b x\right )}{x^{6}}\, dx \]

[In]

integrate(fresnels(b*x)**2/x**6,x)

[Out]

Integral(fresnels(b*x)**2/x**6, x)

Maxima [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^6} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{6}} \,d x } \]

[In]

integrate(fresnel_sin(b*x)^2/x^6,x, algorithm="maxima")

[Out]

integrate(fresnel_sin(b*x)^2/x^6, x)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^6} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{6}} \,d x } \]

[In]

integrate(fresnel_sin(b*x)^2/x^6,x, algorithm="giac")

[Out]

integrate(fresnel_sin(b*x)^2/x^6, x)

Mupad [N/A]

Not integrable

Time = 4.80 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^6} \, dx=\int \frac {{\mathrm {FresnelS}\left (b\,x\right )}^2}{x^6} \,d x \]

[In]

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

[Out]

int(FresnelS(b*x)^2/x^6, x)