Optimal. Leaf size=163 \[ \frac {1}{30} \pi ^3 b^5 S(b x)^2+\frac {\pi b^3}{60 x^2}-\frac {S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {\pi b^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{15 x^3}-\frac {b \sin \left (\pi b^2 x^2\right )}{40 x^4}-\frac {7}{120} \pi ^2 b^5 \text {Si}\left (b^2 \pi x^2\right )+\frac {\pi ^2 b^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{15 x}-\frac {\pi b^3 \cos \left (\pi b^2 x^2\right )}{24 x^2} \]
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Rubi [A] time = 0.21, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6464, 6456, 6440, 30, 3375, 3380, 3297, 3299, 3379} \[ \frac {\pi b^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{15 x^3}+\frac {\pi ^2 b^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{15 x}-\frac {S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{30} \pi ^3 b^5 S(b x)^2-\frac {7}{120} \pi ^2 b^5 \text {Si}\left (b^2 \pi x^2\right )+\frac {\pi b^3}{60 x^2}-\frac {b \sin \left (\pi b^2 x^2\right )}{40 x^4}-\frac {\pi b^3 \cos \left (\pi b^2 x^2\right )}{24 x^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3297
Rule 3299
Rule 3375
Rule 3379
Rule 3380
Rule 6440
Rule 6456
Rule 6464
Rubi steps
\begin {align*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^6} \, dx &=-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 x^5}+\frac {1}{10} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac {1}{5} \left (b^2 \pi \right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=\frac {b^3 \pi }{60 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 x^5}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}+\frac {1}{20} b \operatorname {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{30} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{15} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^2} \, dx\\ &=\frac {b^3 \pi }{60 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 x^5}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{15 x}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{40 x^4}+\frac {1}{60} \left (b^3 \pi \right ) \operatorname {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{40} \left (b^3 \pi \right ) \operatorname {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{30} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx+\frac {1}{15} \left (b^6 \pi ^3\right ) \int S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {b^3 \pi }{60 x^2}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 x^5}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{15 x}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{40 x^4}-\frac {1}{60} b^5 \pi ^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {1}{60} \left (b^5 \pi ^2\right ) \operatorname {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{40} \left (b^5 \pi ^2\right ) \operatorname {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac {1}{15} \left (b^5 \pi ^3\right ) \operatorname {Subst}(\int x \, dx,x,S(b x))\\ &=\frac {b^3 \pi }{60 x^2}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 x^5}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{15 x}+\frac {1}{30} b^5 \pi ^3 S(b x)^2+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{40 x^4}-\frac {7}{120} b^5 \pi ^2 \text {Si}\left (b^2 \pi x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 163, normalized size = 1.00 \[ \frac {1}{30} \pi ^3 b^5 S(b x)^2+\frac {\pi b^3}{60 x^2}-\frac {S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {\pi b^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{15 x^3}-\frac {b \sin \left (\pi b^2 x^2\right )}{40 x^4}-\frac {7}{120} \pi ^2 b^5 \text {Si}\left (b^2 \pi x^2\right )+\frac {\pi ^2 b^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{15 x}-\frac {\pi b^3 \cos \left (\pi b^2 x^2\right )}{24 x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{6}}, x\right ) \]
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 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{x^{6}}\, dx \]
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 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{6}}\,{d x} \]
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 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^6} \,d x \]
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 \[ \int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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