Optimal. Leaf size=270 \[ \frac{1}{384} \pi ^4 b^8 \text{Unintegrable}\left (\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x},x\right )-\frac{\pi ^3 b^6 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{384 x^2}+\frac{\pi b^2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{48 x^6}+\frac{\pi ^2 b^4 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{192 x^4}-\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{8 x^8}+\frac{853 \pi ^4 b^8 S\left (\sqrt{2} b x\right )}{40320 \sqrt{2}}+\frac{\pi b^3}{480 x^5}+\frac{187 \pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{40320 x^3}-\frac{b \sin \left (\pi b^2 x^2\right )}{112 x^7}+\frac{853 \pi ^3 b^7 \cos \left (\pi b^2 x^2\right )}{80640 x}-\frac{19 \pi b^3 \cos \left (\pi b^2 x^2\right )}{3360 x^5}-\frac{\pi ^3 b^7}{768 x} \]
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Rubi [A] time = 0.309526, 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 = {} \[ \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^9} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^9} \, dx &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{8 x^8}+\frac{1}{16} b \int \frac{\sin \left (b^2 \pi x^2\right )}{x^8} \, dx-\frac{1}{8} \left (b^2 \pi \right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^7} \, dx\\ &=\frac{b^3 \pi }{480 x^5}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{8 x^8}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac{b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac{1}{96} \left (b^3 \pi \right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^6} \, dx+\frac{1}{56} \left (b^3 \pi \right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac{1}{48} \left (b^4 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^5} \, dx\\ &=\frac{b^3 \pi }{480 x^5}-\frac{19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{8 x^8}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{192 x^4}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac{b \sin \left (b^2 \pi x^2\right )}{112 x^7}-\frac{1}{384} \left (b^5 \pi ^2\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac{1}{240} \left (b^5 \pi ^2\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac{1}{140} \left (b^5 \pi ^2\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac{1}{192} \left (b^6 \pi ^3\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^3} \, dx\\ &=\frac{b^3 \pi }{480 x^5}-\frac{b^7 \pi ^3}{768 x}-\frac{19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{8 x^8}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{192 x^4}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac{b^6 \pi ^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{384 x^2}-\frac{b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac{187 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{40320 x^3}-\frac{1}{768} \left (b^7 \pi ^3\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{576} \left (b^7 \pi ^3\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{360} \left (b^7 \pi ^3\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{210} \left (b^7 \pi ^3\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac{1}{384} \left (b^8 \pi ^4\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx\\ &=\frac{b^3 \pi }{480 x^5}-\frac{b^7 \pi ^3}{768 x}-\frac{19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}+\frac{853 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{80640 x}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{8 x^8}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{192 x^4}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac{b^6 \pi ^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{384 x^2}-\frac{b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac{187 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{40320 x^3}+\frac{1}{384} \left (b^8 \pi ^4\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx+\frac{1}{384} \left (b^9 \pi ^4\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac{1}{288} \left (b^9 \pi ^4\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac{1}{180} \left (b^9 \pi ^4\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac{1}{105} \left (b^9 \pi ^4\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx\\ &=\frac{b^3 \pi }{480 x^5}-\frac{b^7 \pi ^3}{768 x}-\frac{19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}+\frac{853 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{80640 x}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{8 x^8}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{192 x^4}+\frac{853 b^8 \pi ^4 S\left (\sqrt{2} b x\right )}{40320 \sqrt{2}}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac{b^6 \pi ^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{384 x^2}-\frac{b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac{187 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{40320 x^3}+\frac{1}{384} \left (b^8 \pi ^4\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx\\ \end{align*}
Mathematica [A] time = 0.031928, size = 0, normalized size = 0. \[ \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^9} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{9}}\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 [A] 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^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] 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^{9}}, 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 [A] 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^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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