\(\int \frac {\operatorname {FresnelS}(b x) \sin (\frac {1}{2} b^2 \pi x^2)}{x^3} \, dx\) [82]

Optimal result

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

[Out]

-1/4*b/x+1/4*b*cos(b^2*Pi*x^2)/x-1/2*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^2+1/4*b^2*Pi*FresnelS(b*x*2^(1/2))*2^
(1/2)+1/2*b^2*Pi*Unintegrable(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x,x)

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx=\int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx \]

[In]

Int[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^3,x]

[Out]

-1/4*b/x + (b*Cos[b^2*Pi*x^2])/(4*x) + (b^2*Pi*FresnelS[Sqrt[2]*b*x])/(2*Sqrt[2]) - (FresnelS[b*x]*Sin[(b^2*Pi
*x^2)/2])/(2*x^2) + (b^2*Pi*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x, x])/2

Rubi steps \begin{align*} \text {integral}& = -\frac {b}{4 x}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 x^2}-\frac {1}{4} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{2} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x} \, dx \\ & = -\frac {b}{4 x}+\frac {b \cos \left (b^2 \pi x^2\right )}{4 x}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 x^2}+\frac {1}{2} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x} \, dx+\frac {1}{2} \left (b^3 \pi \right ) \int \sin \left (b^2 \pi x^2\right ) \, dx \\ & = -\frac {b}{4 x}+\frac {b \cos \left (b^2 \pi x^2\right )}{4 x}+\frac {b^2 \pi \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2}}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 x^2}+\frac {1}{2} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx=\int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx \]

[In]

Integrate[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^3,x]

[Out]

Integrate[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^3, x]

Maple [N/A] (verified)

Not integrable

Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \frac {\operatorname {FresnelS}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{3}}d x\]

[In]

int(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^3,x)

[Out]

int(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^3,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{3}} \,d x } \]

[In]

integrate(fresnel_sin(b*x)*sin(1/2*b^2*pi*x^2)/x^3,x, algorithm="fricas")

[Out]

integral(fresnel_sin(b*x)*sin(1/2*pi*b^2*x^2)/x^3, x)

Sympy [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx=\int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{3}}\, dx \]

[In]

integrate(fresnels(b*x)*sin(1/2*b**2*pi*x**2)/x**3,x)

[Out]

Integral(sin(pi*b**2*x**2/2)*fresnels(b*x)/x**3, x)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{3}} \,d x } \]

[In]

integrate(fresnel_sin(b*x)*sin(1/2*b^2*pi*x^2)/x^3,x, algorithm="maxima")

[Out]

integrate(fresnel_sin(b*x)*sin(1/2*pi*b^2*x^2)/x^3, x)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{3}} \,d x } \]

[In]

integrate(fresnel_sin(b*x)*sin(1/2*b^2*pi*x^2)/x^3,x, algorithm="giac")

[Out]

integrate(fresnel_sin(b*x)*sin(1/2*pi*b^2*x^2)/x^3, x)

Mupad [N/A]

Not integrable

Time = 4.84 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^3} \,d x \]

[In]

int((FresnelS(b*x)*sin((Pi*b^2*x^2)/2))/x^3,x)

[Out]

int((FresnelS(b*x)*sin((Pi*b^2*x^2)/2))/x^3, x)