\(\int \frac {\operatorname {FresnelC}(b x)^2}{x^4} \, dx\) [151]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^4} \, dx=-\frac {b^2}{6 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{6 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{3 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{3 x^3}-\frac {b^3 \pi \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{3 \sqrt {2}}-\frac {1}{3} b^3 \pi \text {Int}\left (\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \]

[Out]

-1/6*b^2/x-1/6*b^2*cos(b^2*Pi*x^2)/x-1/3*b*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2-1/3*FresnelC(b*x)^2/x^3-1/6*b
^3*Pi*FresnelS(b*x*2^(1/2))*2^(1/2)-1/3*b^3*Pi*Unintegrable(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x,x)

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 10, 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 = {} \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^4} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^4} \, dx \]

[In]

Int[FresnelC[b*x]^2/x^4,x]

[Out]

-1/6*b^2/x - (b^2*Cos[b^2*Pi*x^2])/(6*x) - (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(3*x^2) - FresnelC[b*x]^2/(3*
x^3) - (b^3*Pi*FresnelS[Sqrt[2]*b*x])/(3*Sqrt[2]) - (b^3*Pi*Defer[Int][(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x,
x])/3

Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x)^2}{3 x^3}+\frac {1}{3} (2 b) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3} \, dx \\ & = -\frac {b^2}{6 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{3 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{3 x^3}+\frac {1}{6} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{3} \left (b^3 \pi \right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ & = -\frac {b^2}{6 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{6 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{3 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{3 x^3}-\frac {1}{3} \left (b^3 \pi \right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx-\frac {1}{3} \left (b^4 \pi \right ) \int \sin \left (b^2 \pi x^2\right ) \, dx \\ & = -\frac {b^2}{6 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{6 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{3 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{3 x^3}-\frac {b^3 \pi \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{3 \sqrt {2}}-\frac {1}{3} \left (b^3 \pi \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 = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^4} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^4} \, dx \]

[In]

Integrate[FresnelC[b*x]^2/x^4,x]

[Out]

Integrate[FresnelC[b*x]^2/x^4, x]

Maple [N/A] (verified)

Not integrable

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {FresnelC}\left (b x \right )^{2}}{x^{4}}d x\]

[In]

int(FresnelC(b*x)^2/x^4,x)

[Out]

int(FresnelC(b*x)^2/x^4,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^4} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{4}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)^2/x^4,x, algorithm="fricas")

[Out]

integral(fresnel_cos(b*x)^2/x^4, x)

Sympy [N/A]

Not integrable

Time = 1.11 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^4} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{4}}\, dx \]

[In]

integrate(fresnelc(b*x)**2/x**4,x)

[Out]

Integral(fresnelc(b*x)**2/x**4, x)

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^4} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{4}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)^2/x^4,x, algorithm="maxima")

[Out]

integrate(fresnel_cos(b*x)^2/x^4, x)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^4} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{4}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)^2/x^4,x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)^2/x^4, x)

Mupad [N/A]

Not integrable

Time = 4.96 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^4} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^4} \,d x \]

[In]

int(FresnelC(b*x)^2/x^4,x)

[Out]

int(FresnelC(b*x)^2/x^4, x)