\(\int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx\) [45]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=-\frac {b^2}{120 x^4}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}+\frac {1}{72} b^6 \pi ^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{45 x^3}-\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^5}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac {1}{45} b^5 \pi ^2 \text {Int}\left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2},x\right ) \]

[Out]

-1/120*b^2/x^4+1/72*b^6*Pi^2*Ci(b^2*Pi*x^2)+1/120*b^2*cos(b^2*Pi*x^2)/x^4-1/45*b^3*Pi*cos(1/2*b^2*Pi*x^2)*Fres
nelS(b*x)/x^3-1/6*FresnelS(b*x)^2/x^6-1/15*b*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^5-1/72*b^4*Pi*sin(b^2*Pi*x^2)
/x^2-1/45*b^5*Pi^2*Unintegrable(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^2,x)

Rubi [N/A]

Not integrable

Time = 0.15 (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 = {} \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx \]

[In]

Int[FresnelS[b*x]^2/x^7,x]

[Out]

-1/120*b^2/x^4 + (b^2*Cos[b^2*Pi*x^2])/(120*x^4) + (b^6*Pi^2*CosIntegral[b^2*Pi*x^2])/72 - (b^3*Pi*Cos[(b^2*Pi
*x^2)/2]*FresnelS[b*x])/(45*x^3) - FresnelS[b*x]^2/(6*x^6) - (b*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(15*x^5) -
(b^4*Pi*Sin[b^2*Pi*x^2])/(72*x^2) - (b^5*Pi^2*Defer[Int][(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x])/45

Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}+\frac {1}{3} b \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6} \, dx \\ & = -\frac {b^2}{120 x^4}-\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^5}-\frac {1}{30} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^5} \, dx+\frac {1}{15} \left (b^3 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^4} \, dx \\ & = -\frac {b^2}{120 x^4}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{45 x^3}-\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^5}-\frac {1}{60} b^2 \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{90} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \\ & = -\frac {b^2}{120 x^4}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{45 x^3}-\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^5}+\frac {1}{180} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{120} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \\ & = -\frac {b^2}{120 x^4}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{45 x^3}-\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^5}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{180} \left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac {1}{120} \left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right ) \\ & = -\frac {b^2}{120 x^4}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}+\frac {1}{72} b^6 \pi ^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{45 x^3}-\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^5}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac {1}{45} \left (b^5 \pi ^2\right ) \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 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx \]

[In]

Integrate[FresnelS[b*x]^2/x^7,x]

[Out]

Integrate[FresnelS[b*x]^2/x^7, x]

Maple [N/A] (verified)

Not integrable

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

\[\int \frac {\operatorname {FresnelS}\left (b x \right )^{2}}{x^{7}}d x\]

[In]

int(FresnelS(b*x)^2/x^7,x)

[Out]

int(FresnelS(b*x)^2/x^7,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{7}} \,d x } \]

[In]

integrate(fresnel_sin(b*x)^2/x^7,x, algorithm="fricas")

[Out]

integral(fresnel_sin(b*x)^2/x^7, x)

Sympy [N/A]

Not integrable

Time = 1.33 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int \frac {S^{2}\left (b x\right )}{x^{7}}\, dx \]

[In]

integrate(fresnels(b*x)**2/x**7,x)

[Out]

Integral(fresnels(b*x)**2/x**7, x)

Maxima [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{7}} \,d x } \]

[In]

integrate(fresnel_sin(b*x)^2/x^7,x, algorithm="maxima")

[Out]

integrate(fresnel_sin(b*x)^2/x^7, x)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{7}} \,d x } \]

[In]

integrate(fresnel_sin(b*x)^2/x^7,x, algorithm="giac")

[Out]

integrate(fresnel_sin(b*x)^2/x^7, x)

Mupad [N/A]

Not integrable

Time = 4.85 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int \frac {{\mathrm {FresnelS}\left (b\,x\right )}^2}{x^7} \,d x \]

[In]

int(FresnelS(b*x)^2/x^7,x)

[Out]

int(FresnelS(b*x)^2/x^7, x)