Optimal. Leaf size=163 \[ \frac{\pi b^2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{15 x^3}+\frac{\pi ^2 b^4 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{15 x}-\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac{1}{30} \pi ^3 b^5 S(b x)^2-\frac{7}{120} \pi ^2 b^5 \text{Si}\left (b^2 \pi x^2\right )+\frac{\pi b^3}{60 x^2}-\frac{b \sin \left (\pi b^2 x^2\right )}{40 x^4}-\frac{\pi b^3 \cos \left (\pi b^2 x^2\right )}{24 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.213637, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6464, 6456, 6440, 30, 3375, 3380, 3297, 3299, 3379} \[ \frac{\pi b^2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{15 x^3}+\frac{\pi ^2 b^4 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{15 x}-\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac{1}{30} \pi ^3 b^5 S(b x)^2-\frac{7}{120} \pi ^2 b^5 \text{Si}\left (b^2 \pi x^2\right )+\frac{\pi b^3}{60 x^2}-\frac{b \sin \left (\pi b^2 x^2\right )}{40 x^4}-\frac{\pi b^3 \cos \left (\pi b^2 x^2\right )}{24 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6464
Rule 6456
Rule 6440
Rule 30
Rule 3375
Rule 3380
Rule 3297
Rule 3299
Rule 3379
Rubi steps
\begin{align*} \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^6} \, dx &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{5 x^5}+\frac{1}{10} b \int \frac{\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac{1}{5} \left (b^2 \pi \right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=\frac{b^3 \pi }{60 x^2}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{5 x^5}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{15 x^3}+\frac{1}{20} b \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac{1}{30} \left (b^3 \pi \right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac{1}{15} \left (b^4 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^2} \, dx\\ &=\frac{b^3 \pi }{60 x^2}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{5 x^5}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{15 x}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{40 x^4}+\frac{1}{60} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac{1}{40} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{30} \left (b^5 \pi ^2\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x} \, dx+\frac{1}{15} \left (b^6 \pi ^3\right ) \int S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac{b^3 \pi }{60 x^2}-\frac{b^3 \pi \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{5 x^5}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{15 x}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{40 x^4}-\frac{1}{60} b^5 \pi ^2 \text{Si}\left (b^2 \pi x^2\right )-\frac{1}{60} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac{1}{40} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac{1}{15} \left (b^5 \pi ^3\right ) \operatorname{Subst}(\int x \, dx,x,S(b x))\\ &=\frac{b^3 \pi }{60 x^2}-\frac{b^3 \pi \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{5 x^5}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{15 x}+\frac{1}{30} b^5 \pi ^3 S(b x)^2+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{40 x^4}-\frac{7}{120} b^5 \pi ^2 \text{Si}\left (b^2 \pi x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0100253, size = 163, normalized size = 1. \[ \frac{\pi b^2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{15 x^3}+\frac{\pi ^2 b^4 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{15 x}-\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac{1}{30} \pi ^3 b^5 S(b x)^2-\frac{7}{120} \pi ^2 b^5 \text{Si}\left (b^2 \pi x^2\right )+\frac{\pi b^3}{60 x^2}-\frac{b \sin \left (\pi b^2 x^2\right )}{40 x^4}-\frac{\pi b^3 \cos \left (\pi b^2 x^2\right )}{24 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{6}}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )}{x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]