Optimal. Leaf size=259 \[ -\frac {1}{168} \pi ^3 b^7 \text {Int}\left (\frac {S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x},x\right )-\frac {2}{315} \sqrt {2} \pi ^3 b^7 S\left (\sqrt {2} b x\right )-\frac {\pi ^3 b^7 S\left (\sqrt {2} b x\right )}{72 \sqrt {2}}+\frac {\pi ^2 b^6}{336 x}-\frac {b S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{21 x^6}-\frac {b^2}{210 x^5}+\frac {b^2 \cos \left (\pi b^2 x^2\right )}{210 x^5}-\frac {67 \pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{5040 x}+\frac {\pi ^2 b^5 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{168 x^2}-\frac {13 \pi b^4 \sin \left (\pi b^2 x^2\right )}{2520 x^3}-\frac {\pi b^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{84 x^4}-\frac {S(b x)^2}{7 x^7} \]
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Rubi [A] time = 0.24, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {S(b x)^2}{x^8} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {S(b x)^2}{x^8} \, dx &=-\frac {S(b x)^2}{7 x^7}+\frac {1}{7} (2 b) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^7} \, dx\\ &=-\frac {b^2}{210 x^5}-\frac {S(b x)^2}{7 x^7}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{21 x^6}-\frac {1}{42} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6} \, dx+\frac {1}{21} \left (b^3 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^5} \, dx\\ &=-\frac {b^2}{210 x^5}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{84 x^4}-\frac {S(b x)^2}{7 x^7}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{21 x^6}+\frac {1}{168} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac {1}{105} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{84} \left (b^5 \pi ^2\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx\\ &=-\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{84 x^4}-\frac {S(b x)^2}{7 x^7}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{21 x^6}+\frac {b^5 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^2}-\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac {1}{336} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{252} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{315} \left (2 b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{168} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx\\ &=-\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac {67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{84 x^4}-\frac {S(b x)^2}{7 x^7}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{21 x^6}+\frac {b^5 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^2}-\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}-\frac {1}{168} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx-\frac {1}{168} \left (b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx-\frac {1}{126} \left (b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx-\frac {1}{315} \left (4 b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac {67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{84 x^4}-\frac {S(b x)^2}{7 x^7}-\frac {b^7 \pi ^3 S\left (\sqrt {2} b x\right )}{72 \sqrt {2}}-\frac {2}{315} \sqrt {2} b^7 \pi ^3 S\left (\sqrt {2} b x\right )-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{21 x^6}+\frac {b^5 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^2}-\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}-\frac {1}{168} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {S(b x)^2}{x^8} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right )^{2}}{x^{8}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnels}\left (b x\right )^{2}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {S}\left (b x \right )^{2}}{x^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnels}\left (b x\right )^{2}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {FresnelS}\left (b\,x\right )}^2}{x^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {S^{2}\left (b x\right )}{x^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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