\(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelC}(b x)}{x^5} \, dx\) [193]

Optimal result

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

[Out]

-1/24*b/x^3-1/24*b*cos(b^2*Pi*x^2)/x^3-1/4*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^4+1/8*b^2*Pi*FresnelC(b*x)*sin(
1/2*b^2*Pi*x^2)/x^2+7/48*b^3*Pi*sin(b^2*Pi*x^2)/x-7/48*b^4*Pi^2*FresnelC(b*x*2^(1/2))*2^(1/2)-1/8*b^4*Pi^2*Uni
ntegrable(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x,x)

Rubi [N/A]

Not integrable

Time = 0.09 (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 = {} \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^5} \, dx=\int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^5} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^5} \, dx=\int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^5} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^5} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{5}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^5} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{5}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.64 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^5} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^5} \,d x \]

[In]

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

[Out]

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