\(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelS}(b x)}{x^4} \, dx\) [103]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^4} \, dx=\frac {1}{12} b^3 \pi \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{12 x^2}-\frac {1}{3} b^2 \pi \text {Int}\left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^4} \, dx=\int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^4} \, dx \]

[In]

Int[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^4,x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{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 {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \\ & = -\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{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 {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \\ & = -\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{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 {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{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 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{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 {\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 = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^4} \, dx=\int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^4} \, dx \]

[In]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^4,x]

[Out]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^4, x]

Maple [N/A] (verified)

Not integrable

Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelS}\left (b x \right )}{x^{4}}d x\]

[In]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^4,x)

[Out]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^4,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^4} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{4}} \,d x } \]

[In]

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

[Out]

integral(cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x)/x^4, x)

Sympy [N/A]

Not integrable

Time = 1.68 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^4} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{4}}\, dx \]

[In]

integrate(cos(1/2*b**2*pi*x**2)*fresnels(b*x)/x**4,x)

[Out]

Integral(cos(pi*b**2*x**2/2)*fresnels(b*x)/x**4, x)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^4} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{4}} \,d x } \]

[In]

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

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x)/x^4, x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^4} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{4}} \,d x } \]

[In]

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

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x)/x^4, x)

Mupad [N/A]

Not integrable

Time = 4.75 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^4} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^4} \,d x \]

[In]

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^4,x)

[Out]

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^4, x)