Optimal. Leaf size=165 \[ -\frac{1}{45} \pi ^2 b^5 \text{Unintegrable}\left (\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x^2},x\right )+\frac{1}{72} \pi ^2 b^6 \text{CosIntegral}\left (\pi b^2 x^2\right )-\frac{b S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{15 x^5}-\frac{\pi b^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{45 x^3}-\frac{b^2}{120 x^4}-\frac{\pi b^4 \sin \left (\pi b^2 x^2\right )}{72 x^2}+\frac{b^2 \cos \left (\pi b^2 x^2\right )}{120 x^4}-\frac{S(b x)^2}{6 x^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.222055, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{S(b x)^2}{x^7} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{S(b x)^2}{x^7} \, dx &=-\frac{S(b x)^2}{6 x^6}+\frac{1}{3} b \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^6} \, dx\\ &=-\frac{b^2}{120 x^4}-\frac{S(b x)^2}{6 x^6}-\frac{b S(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 ) S(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 ) S(b x)}{45 x^3}-\frac{S(b x)^2}{6 x^6}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{15 x^5}-\frac{1}{60} b^2 \operatorname{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{S(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 ) S(b x)}{45 x^3}-\frac{S(b x)^2}{6 x^6}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{15 x^5}+\frac{1}{180} \left (b^4 \pi \right ) \operatorname{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 ) \operatorname{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{S(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 ) S(b x)}{45 x^3}-\frac{S(b x)^2}{6 x^6}-\frac{b S(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{S(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 ) \operatorname{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 ) \operatorname{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 \text{Ci}\left (b^2 \pi x^2\right )-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{45 x^3}-\frac{S(b x)^2}{6 x^6}-\frac{b S(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{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ \end{align*}
Mathematica [A] time = 0.0202359, size = 0, normalized size = 0. \[ \int \frac{S(b x)^2}{x^7} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it FresnelS} \left ( bx \right ) \right ) ^{2}}{{x}^{7}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right )^{2}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnels}\left (b x\right )^{2}}{x^{7}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{S^{2}\left (b x\right )}{x^{7}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right )^{2}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]