Optimal. Leaf size=278 \[ -\frac{\pi ^3 b^6 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x^3}+\frac{\pi b^2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{63 x^7}-\frac{\pi ^4 b^8 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x}+\frac{\pi ^2 b^4 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{315 x^5}-\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac{\pi ^5 b^9 S(b x)^2}{1890}+\frac{83 \pi ^4 b^9 \text{Si}\left (b^2 \pi x^2\right )}{30240}-\frac{\pi ^3 b^7}{3780 x^2}+\frac{\pi b^3}{756 x^6}+\frac{67 \pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{30240 x^4}-\frac{b \sin \left (\pi b^2 x^2\right )}{144 x^8}+\frac{5 \pi ^3 b^7 \cos \left (\pi b^2 x^2\right )}{2016 x^2}-\frac{11 \pi b^3 \cos \left (\pi b^2 x^2\right )}{3024 x^6} \]
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Rubi [A] time = 0.516173, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 26, 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} \[ -\frac{\pi ^3 b^6 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x^3}+\frac{\pi b^2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{63 x^7}-\frac{\pi ^4 b^8 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x}+\frac{\pi ^2 b^4 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{315 x^5}-\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac{\pi ^5 b^9 S(b x)^2}{1890}+\frac{83 \pi ^4 b^9 \text{Si}\left (b^2 \pi x^2\right )}{30240}-\frac{\pi ^3 b^7}{3780 x^2}+\frac{\pi b^3}{756 x^6}+\frac{67 \pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{30240 x^4}-\frac{b \sin \left (\pi b^2 x^2\right )}{144 x^8}+\frac{5 \pi ^3 b^7 \cos \left (\pi b^2 x^2\right )}{2016 x^2}-\frac{11 \pi b^3 \cos \left (\pi b^2 x^2\right )}{3024 x^6} \]
Antiderivative was successfully verified.
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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^{10}} \, dx &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac{1}{18} b \int \frac{\sin \left (b^2 \pi x^2\right )}{x^9} \, dx-\frac{1}{9} \left (b^2 \pi \right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^8} \, dx\\ &=\frac{b^3 \pi }{756 x^6}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{63 x^7}+\frac{1}{36} b \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^5} \, dx,x,x^2\right )+\frac{1}{126} \left (b^3 \pi \right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^7} \, dx-\frac{1}{63} \left (b^4 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^6} \, dx\\ &=\frac{b^3 \pi }{756 x^6}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{315 x^5}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac{b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac{1}{252} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac{1}{144} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )-\frac{1}{630} \left (b^5 \pi ^2\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^5} \, dx+\frac{1}{315} \left (b^6 \pi ^3\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=\frac{b^3 \pi }{756 x^6}-\frac{b^7 \pi ^3}{3780 x^2}-\frac{11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{315 x^5}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac{b^6 \pi ^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{945 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{144 x^8}-\frac{\left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )}{1260}-\frac{1}{756} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac{1}{432} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac{\left (b^7 \pi ^3\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^3} \, dx}{1890}+\frac{1}{945} \left (b^8 \pi ^4\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^2} \, dx\\ &=\frac{b^3 \pi }{756 x^6}-\frac{b^7 \pi ^3}{3780 x^2}-\frac{11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{315 x^5}-\frac{b^8 \pi ^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{945 x}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac{b^6 \pi ^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{945 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac{67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}-\frac{\left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{3780}-\frac{\left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{2520}-\frac{\left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1512}-\frac{1}{864} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac{\left (b^9 \pi ^4\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x} \, dx}{1890}-\frac{1}{945} \left (b^{10} \pi ^5\right ) \int S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac{b^3 \pi }{756 x^6}-\frac{b^7 \pi ^3}{3780 x^2}-\frac{11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}+\frac{5 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{2016 x^2}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{315 x^5}-\frac{b^8 \pi ^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{945 x}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac{b^6 \pi ^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{945 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac{67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}+\frac{b^9 \pi ^4 \text{Si}\left (b^2 \pi x^2\right )}{3780}+\frac{\left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{3780}+\frac{\left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{2520}+\frac{\left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1512}+\frac{1}{864} \left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac{1}{945} \left (b^9 \pi ^5\right ) \operatorname{Subst}(\int x \, dx,x,S(b x))\\ &=\frac{b^3 \pi }{756 x^6}-\frac{b^7 \pi ^3}{3780 x^2}-\frac{11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}+\frac{5 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{2016 x^2}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{315 x^5}-\frac{b^8 \pi ^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{945 x}-\frac{b^9 \pi ^5 S(b x)^2}{1890}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac{b^6 \pi ^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{945 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac{67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}+\frac{83 b^9 \pi ^4 \text{Si}\left (b^2 \pi x^2\right )}{30240}\\ \end{align*}
Mathematica [A] time = 0.01589, size = 278, normalized size = 1. \[ -\frac{\pi ^3 b^6 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x^3}+\frac{\pi b^2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{63 x^7}-\frac{\pi ^4 b^8 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x}+\frac{\pi ^2 b^4 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{315 x^5}-\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac{\pi ^5 b^9 S(b x)^2}{1890}+\frac{83 \pi ^4 b^9 \text{Si}\left (b^2 \pi x^2\right )}{30240}-\frac{\pi ^3 b^7}{3780 x^2}+\frac{\pi b^3}{756 x^6}+\frac{67 \pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{30240 x^4}-\frac{b \sin \left (\pi b^2 x^2\right )}{144 x^8}+\frac{5 \pi ^3 b^7 \cos \left (\pi b^2 x^2\right )}{2016 x^2}-\frac{11 \pi b^3 \cos \left (\pi b^2 x^2\right )}{3024 x^6} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{10}}\cos \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{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )}{x^{10}}\,{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{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )}{x^{10}}, 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{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )}{x^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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