\(\int \frac {\operatorname {FresnelC}(b x) \sin (\frac {1}{2} b^2 \pi x^2)}{x^5} \, dx\) [213]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx=-\frac {b^3 \pi }{16 x}-\frac {7 b^3 \pi \cos \left (b^2 \pi x^2\right )}{48 x}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{8 x^2}-\frac {7 b^4 \pi ^2 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{24 \sqrt {2}}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{24 x^3}-\frac {1}{8} b^4 \pi ^2 \text {Int}\left (\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \]

[Out]

-1/16*b^3*Pi/x-7/48*b^3*Pi*cos(b^2*Pi*x^2)/x-1/8*b^2*Pi*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2-1/4*FresnelC(b*x
)*sin(1/2*b^2*Pi*x^2)/x^4-1/24*b*sin(b^2*Pi*x^2)/x^3-7/48*b^4*Pi^2*FresnelS(b*x*2^(1/2))*2^(1/2)-1/8*b^4*Pi^2*
Unintegrable(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x,x)

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx=\int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx \]

[In]

Int[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^5,x]

[Out]

-1/16*(b^3*Pi)/x - (7*b^3*Pi*Cos[b^2*Pi*x^2])/(48*x) - (b^2*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(8*x^2) - (7
*b^4*Pi^2*FresnelS[Sqrt[2]*b*x])/(24*Sqrt[2]) - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(4*x^4) - (b*Sin[b^2*Pi*x^
2])/(24*x^3) - (b^4*Pi^2*Defer[Int][(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/8

Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 x^4}+\frac {1}{8} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac {1}{4} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3} \, dx \\ & = -\frac {b^3 \pi }{16 x}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{8 x^2}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{24 x^3}+\frac {1}{16} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{12} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{8} \left (b^4 \pi ^2\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ & = -\frac {b^3 \pi }{16 x}-\frac {7 b^3 \pi \cos \left (b^2 \pi x^2\right )}{48 x}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{8 x^2}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{24 x^3}-\frac {1}{8} \left (b^4 \pi ^2\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx-\frac {1}{8} \left (b^5 \pi ^2\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx-\frac {1}{6} \left (b^5 \pi ^2\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx \\ & = -\frac {b^3 \pi }{16 x}-\frac {7 b^3 \pi \cos \left (b^2 \pi x^2\right )}{48 x}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{8 x^2}-\frac {7 b^4 \pi ^2 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{24 \sqrt {2}}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{24 x^3}-\frac {1}{8} \left (b^4 \pi ^2\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx=\int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx \]

[In]

Integrate[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^5,x]

[Out]

Integrate[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^5, x]

Maple [N/A] (verified)

Not integrable

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

\[\int \frac {\operatorname {FresnelC}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{5}}d x\]

[In]

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^5,x)

[Out]

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^5,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{5}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2)/x^5,x, algorithm="fricas")

[Out]

integral(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2)/x^5, x)

Sympy [N/A]

Not integrable

Time = 3.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx=\int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{5}}\, dx \]

[In]

integrate(fresnelc(b*x)*sin(1/2*b**2*pi*x**2)/x**5,x)

[Out]

Integral(sin(pi*b**2*x**2/2)*fresnelc(b*x)/x**5, x)

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{5}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2)/x^5,x, algorithm="maxima")

[Out]

integrate(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2)/x^5, x)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{5}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2)/x^5,x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2)/x^5, x)

Mupad [N/A]

Not integrable

Time = 4.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^5} \,d x \]

[In]

int((FresnelC(b*x)*sin((Pi*b^2*x^2)/2))/x^5,x)

[Out]

int((FresnelC(b*x)*sin((Pi*b^2*x^2)/2))/x^5, x)