Optimal. Leaf size=127 \[ -\frac {1}{12} \pi ^2 b^4 S(b x)^2-\frac {b S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^3}-\frac {b^2}{24 x^2}+\frac {b^2 \cos \left (\pi b^2 x^2\right )}{24 x^2}+\frac {1}{12} \pi b^4 \text {Si}\left (b^2 \pi x^2\right )-\frac {\pi b^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x}-\frac {S(b x)^2}{4 x^4} \]
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Rubi [A] time = 0.14, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6430, 6456, 6464, 6440, 30, 3375, 3380, 3297, 3299} \[ -\frac {b S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^3}-\frac {\pi b^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x}-\frac {1}{12} \pi ^2 b^4 S(b x)^2+\frac {1}{12} \pi b^4 \text {Si}\left (b^2 \pi x^2\right )-\frac {b^2}{24 x^2}+\frac {b^2 \cos \left (\pi b^2 x^2\right )}{24 x^2}-\frac {S(b x)^2}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3297
Rule 3299
Rule 3375
Rule 3380
Rule 6430
Rule 6440
Rule 6456
Rule 6464
Rubi steps
\begin {align*} \int \frac {S(b x)^2}{x^5} \, dx &=-\frac {S(b x)^2}{4 x^4}+\frac {1}{2} b \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=-\frac {b^2}{24 x^2}-\frac {S(b x)^2}{4 x^4}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x^3}-\frac {1}{12} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx+\frac {1}{6} \left (b^3 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^2} \, dx\\ &=-\frac {b^2}{24 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x}-\frac {S(b x)^2}{4 x^4}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x^3}-\frac {1}{24} b^2 \operatorname {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{12} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx-\frac {1}{6} \left (b^5 \pi ^2\right ) \int S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac {b^2}{24 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x}-\frac {S(b x)^2}{4 x^4}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x^3}+\frac {1}{24} b^4 \pi \text {Si}\left (b^2 \pi x^2\right )+\frac {1}{24} \left (b^4 \pi \right ) \operatorname {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{6} \left (b^4 \pi ^2\right ) \operatorname {Subst}(\int x \, dx,x,S(b x))\\ &=-\frac {b^2}{24 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x}-\frac {1}{12} b^4 \pi ^2 S(b x)^2-\frac {S(b x)^2}{4 x^4}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x^3}+\frac {1}{12} b^4 \pi \text {Si}\left (b^2 \pi x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 127, normalized size = 1.00 \[ -\frac {1}{12} \pi ^2 b^4 S(b x)^2-\frac {b S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^3}-\frac {b^2}{24 x^2}+\frac {b^2 \cos \left (\pi b^2 x^2\right )}{24 x^2}+\frac {1}{12} \pi b^4 \text {Si}\left (b^2 \pi x^2\right )-\frac {\pi b^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x}-\frac {S(b x)^2}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right )^{2}}{x^{5}}, x\right ) \]
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 \[ \int \frac {{\rm fresnels}\left (b x\right )^{2}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {S}\left (b x \right )^{2}}{x^{5}}\, dx \]
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 \[ \int \frac {{\rm fresnels}\left (b x\right )^{2}}{x^{5}}\,{d x} \]
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 \[ \int \frac {{\mathrm {FresnelS}\left (b\,x\right )}^2}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {S^{2}\left (b x\right )}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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