Optimal. Leaf size=224 \[ \frac{\pi ^2 b^4 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 x^7}+\frac{\pi ^3 b^6 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x}-\frac{\pi b^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{35 x^5}+\frac{1}{210} \pi ^4 b^7 S(b x)^2-\frac{1}{70} \pi ^3 b^7 \text{Si}\left (b^2 \pi x^2\right )+\frac{\pi ^2 b^5}{420 x^2}-\frac{\pi b^3 \sin \left (\pi b^2 x^2\right )}{105 x^4}-\frac{\pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{84 x^2}+\frac{b \cos \left (\pi b^2 x^2\right )}{84 x^6}-\frac{b}{84 x^6} \]
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Rubi [A] time = 0.353489, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6456, 6464, 6440, 30, 3375, 3380, 3297, 3299, 3379} \[ \frac{\pi ^2 b^4 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 x^7}+\frac{\pi ^3 b^6 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x}-\frac{\pi b^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{35 x^5}+\frac{1}{210} \pi ^4 b^7 S(b x)^2-\frac{1}{70} \pi ^3 b^7 \text{Si}\left (b^2 \pi x^2\right )+\frac{\pi ^2 b^5}{420 x^2}-\frac{\pi b^3 \sin \left (\pi b^2 x^2\right )}{105 x^4}-\frac{\pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{84 x^2}+\frac{b \cos \left (\pi b^2 x^2\right )}{84 x^6}-\frac{b}{84 x^6} \]
Antiderivative was successfully verified.
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Rule 6456
Rule 6464
Rule 6440
Rule 30
Rule 3375
Rule 3380
Rule 3297
Rule 3299
Rule 3379
Rubi steps
\begin{align*} \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^8} \, dx &=-\frac{b}{84 x^6}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 x^7}-\frac{1}{14} b \int \frac{\cos \left (b^2 \pi x^2\right )}{x^7} \, dx+\frac{1}{7} \left (b^2 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^6} \, dx\\ &=-\frac{b}{84 x^6}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{35 x^5}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 x^7}-\frac{1}{28} b \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac{1}{70} \left (b^3 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac{1}{35} \left (b^4 \pi ^2\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=-\frac{b}{84 x^6}+\frac{b^5 \pi ^2}{420 x^2}+\frac{b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{35 x^5}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{105 x^3}+\frac{1}{140} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac{1}{84} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac{1}{210} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac{1}{105} \left (b^6 \pi ^3\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^2} \, dx\\ &=-\frac{b}{84 x^6}+\frac{b^5 \pi ^2}{420 x^2}+\frac{b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{35 x^5}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{105 x}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac{b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}+\frac{1}{420} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac{1}{280} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac{1}{168} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{210} \left (b^7 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x} \, dx+\frac{1}{105} \left (b^8 \pi ^4\right ) \int S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac{b}{84 x^6}+\frac{b^5 \pi ^2}{420 x^2}+\frac{b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac{b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{35 x^5}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{105 x}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac{b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac{1}{420} b^7 \pi ^3 \text{Si}\left (b^2 \pi x^2\right )-\frac{1}{420} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac{1}{280} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac{1}{168} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac{1}{105} \left (b^7 \pi ^4\right ) \operatorname{Subst}(\int x \, dx,x,S(b x))\\ &=-\frac{b}{84 x^6}+\frac{b^5 \pi ^2}{420 x^2}+\frac{b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac{b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{35 x^5}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{105 x}+\frac{1}{210} b^7 \pi ^4 S(b x)^2-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac{b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac{1}{70} b^7 \pi ^3 \text{Si}\left (b^2 \pi x^2\right )\\ \end{align*}
Mathematica [A] time = 0.012694, size = 224, normalized size = 1. \[ \frac{\pi ^2 b^4 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 x^7}+\frac{\pi ^3 b^6 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x}-\frac{\pi b^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{35 x^5}+\frac{1}{210} \pi ^4 b^7 S(b x)^2-\frac{1}{70} \pi ^3 b^7 \text{Si}\left (b^2 \pi x^2\right )+\frac{\pi ^2 b^5}{420 x^2}-\frac{\pi b^3 \sin \left (\pi b^2 x^2\right )}{105 x^4}-\frac{\pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{84 x^2}+\frac{b \cos \left (\pi b^2 x^2\right )}{84 x^6}-\frac{b}{84 x^6} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{8}}\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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