\(\int \frac {\operatorname {FresnelC}(b x)^2}{x^2} \, dx\) [149]

Optimal result

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

[Out]

-FresnelC(b*x)^2/x+2*b*Unintegrable(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x,x)

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^2} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^2} \, dx \]

[In]

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

[Out]

-(FresnelC[b*x]^2/x) + 2*b*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x)^2}{x}+(2 b) \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 = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^2} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^2} \, dx \]

[In]

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

[Out]

Integrate[FresnelC[b*x]^2/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

integral(fresnel_cos(b*x)^2/x^2, x)

Sympy [N/A]

Not integrable

Time = 1.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^2} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{2}}\, dx \]

[In]

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

[Out]

Integral(fresnelc(b*x)**2/x**2, x)

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

integrate(fresnel_cos(b*x)^2/x^2, x)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^2} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(fresnel_cos(b*x)^2/x^2, x)

Mupad [N/A]

Not integrable

Time = 4.91 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^2} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^2} \,d x \]

[In]

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

[Out]

int(FresnelC(b*x)^2/x^2, x)