\(\int \frac {\operatorname {FresnelS}(b x)^2}{x^3} \, dx\) [41]

Optimal result

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

[Out]

-1/2*FresnelS(b*x)^2/x^2+b*Unintegrable(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^2,x)

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^3} \, dx=\int \frac {\operatorname {FresnelS}(b x)^2}{x^3} \, dx \]

[In]

Int[FresnelS[b*x]^2/x^3,x]

[Out]

-1/2*FresnelS[b*x]^2/x^2 + b*Defer[Int][(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelS}(b x)^2}{2 x^2}+b \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 = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^3} \, dx=\int \frac {\operatorname {FresnelS}(b x)^2}{x^3} \, dx \]

[In]

Integrate[FresnelS[b*x]^2/x^3,x]

[Out]

Integrate[FresnelS[b*x]^2/x^3, x]

Maple [N/A] (verified)

Not integrable

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

\[\int \frac {\operatorname {FresnelS}\left (b x \right )^{2}}{x^{3}}d x\]

[In]

int(FresnelS(b*x)^2/x^3,x)

[Out]

int(FresnelS(b*x)^2/x^3,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^3} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{3}} \,d x } \]

[In]

integrate(fresnel_sin(b*x)^2/x^3,x, algorithm="fricas")

[Out]

integral(fresnel_sin(b*x)^2/x^3, x)

Sympy [N/A]

Not integrable

Time = 1.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^3} \, dx=\int \frac {S^{2}\left (b x\right )}{x^{3}}\, dx \]

[In]

integrate(fresnels(b*x)**2/x**3,x)

[Out]

Integral(fresnels(b*x)**2/x**3, x)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^3} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{3}} \,d x } \]

[In]

integrate(fresnel_sin(b*x)^2/x^3,x, algorithm="maxima")

[Out]

integrate(fresnel_sin(b*x)^2/x^3, x)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^3} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{3}} \,d x } \]

[In]

integrate(fresnel_sin(b*x)^2/x^3,x, algorithm="giac")

[Out]

integrate(fresnel_sin(b*x)^2/x^3, x)

Mupad [N/A]

Not integrable

Time = 4.83 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^3} \, dx=\int \frac {{\mathrm {FresnelS}\left (b\,x\right )}^2}{x^3} \,d x \]

[In]

int(FresnelS(b*x)^2/x^3,x)

[Out]

int(FresnelS(b*x)^2/x^3, x)