Optimal. Leaf size=69 \[ -\frac {1}{12} \pi ^2 b^4 S(b x)-\frac {b \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{12 x^3}-\frac {\pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{12 x}-\frac {S(b x)}{4 x^4} \]
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Rubi [A] time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6426, 3387, 3388, 3351} \[ -\frac {1}{12} \pi ^2 b^4 S(b x)-\frac {b \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{12 x^3}-\frac {\pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{12 x}-\frac {S(b x)}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3387
Rule 3388
Rule 6426
Rubi steps
\begin {align*} \int \frac {S(b x)}{x^5} \, dx &=-\frac {S(b x)}{4 x^4}+\frac {1}{4} b \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=-\frac {S(b x)}{4 x^4}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3}+\frac {1}{12} \left (b^3 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x}-\frac {S(b x)}{4 x^4}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3}-\frac {1}{12} \left (b^5 \pi ^2\right ) \int \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x}-\frac {1}{12} b^4 \pi ^2 S(b x)-\frac {S(b x)}{4 x^4}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 69, normalized size = 1.00 \[ -\frac {1}{12} \pi ^2 b^4 S(b x)-\frac {b \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{12 x^3}-\frac {\pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{12 x}-\frac {S(b x)}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right )}{x^{5}}, x\right ) \]
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 \[ \int \frac {{\rm fresnels}\left (b x\right )}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 65, normalized size = 0.94 \[ b^{4} \left (-\frac {\mathrm {S}\left (b x \right )}{4 b^{4} x^{4}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{12 b^{3} x^{3}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}-\pi \,\mathrm {S}\left (b x \right )\right )}{12}\right ) \]
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 \[ \int \frac {{\rm fresnels}\left (b x\right )}{x^{5}}\,{d x} \]
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 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.96, size = 110, normalized size = 1.59 \[ \frac {\pi ^{2} b^{4} S\left (b x\right ) \Gamma \left (- \frac {1}{4}\right )}{64 \Gamma \left (\frac {7}{4}\right )} + \frac {\pi b^{3} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {1}{4}\right )}{64 x \Gamma \left (\frac {7}{4}\right )} + \frac {b \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {1}{4}\right )}{64 x^{3} \Gamma \left (\frac {7}{4}\right )} + \frac {3 S\left (b x\right ) \Gamma \left (- \frac {1}{4}\right )}{64 x^{4} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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