\(\int \frac {\operatorname {FresnelS}(b x) \sin (\frac {1}{2} b^2 \pi x^2)}{x^2} \, dx\) [81]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx=\text {Int}\left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx=\int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx=\int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx=\int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.88 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^2} \,d x \]

[In]

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

[Out]

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