Optimal. Leaf size=102 \[ -\frac {b \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{42 x^6}-\frac {1}{672} \pi ^3 b^7 \text {Ci}\left (\frac {1}{2} b^2 \pi x^2\right )+\frac {\pi ^2 b^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{336 x^2}-\frac {\pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{168 x^4}-\frac {S(b x)}{7 x^7} \]
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Rubi [A] time = 0.12, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6426, 3379, 3297, 3302} \[ -\frac {1}{672} \pi ^3 b^7 \text {CosIntegral}\left (\frac {1}{2} \pi b^2 x^2\right )+\frac {\pi ^2 b^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{336 x^2}-\frac {b \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{42 x^6}-\frac {\pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{168 x^4}-\frac {S(b x)}{7 x^7} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3302
Rule 3379
Rule 6426
Rubi steps
\begin {align*} \int \frac {S(b x)}{x^8} \, dx &=-\frac {S(b x)}{7 x^7}+\frac {1}{7} b \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^7} \, dx\\ &=-\frac {S(b x)}{7 x^7}+\frac {1}{14} b \operatorname {Subst}\left (\int \frac {\sin \left (\frac {1}{2} b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )\\ &=-\frac {S(b x)}{7 x^7}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}+\frac {1}{84} \left (b^3 \pi \right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {1}{2} b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )\\ &=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^4}-\frac {S(b x)}{7 x^7}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}-\frac {1}{336} \left (b^5 \pi ^2\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {1}{2} b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )\\ &=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^4}-\frac {S(b x)}{7 x^7}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{336 x^2}-\frac {1}{672} \left (b^7 \pi ^3\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {1}{2} b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^4}-\frac {1}{672} b^7 \pi ^3 \text {Ci}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {S(b x)}{7 x^7}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{336 x^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 85, normalized size = 0.83 \[ \frac {1}{672} \left (-\pi ^3 b^7 \text {Ci}\left (\frac {1}{2} b^2 \pi x^2\right )+\frac {2 b \left (\pi ^2 b^4 x^4-8\right ) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x^6}-\frac {4 \pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x^4}-\frac {96 S(b x)}{x^7}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right )}{x^{8}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnels}\left (b x\right )}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 93, normalized size = 0.91 \[ b^{7} \left (-\frac {\mathrm {S}\left (b x \right )}{7 b^{7} x^{7}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{42 b^{6} x^{6}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 b^{4} x^{4}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b^{2} x^{2}}+\frac {\pi \Ci \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{4}\right )}{42}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnels}\left (b x\right )}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )}{x^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.05, size = 68, normalized size = 0.67 \[ \frac {\pi ^{5} b^{11} x^{4} \Gamma \left (\frac {11}{4}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {11}{4} \\ 2, 3, \frac {7}{2}, \frac {15}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{61440 \Gamma \left (\frac {15}{4}\right )} - \frac {\pi ^{3} b^{7} \log {\left (b^{4} x^{4} \right )}}{1344} - \frac {\pi b^{3}}{24 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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