Optimal. Leaf size=59 \[ \frac {C\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^2}+\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {x}{2 \pi b} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6460, 3357, 3352} \[ \frac {\text {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^2}+\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {x}{2 \pi b} \]
Antiderivative was successfully verified.
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Rule 3352
Rule 3357
Rule 6460
Rubi steps
\begin {align*} \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx &=\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {\int \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {\int \left (\frac {1}{2}-\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{b \pi }\\ &=-\frac {x}{2 b \pi }+\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {\int \cos \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac {x}{2 b \pi }+\frac {C\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2 \pi }+\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }\\ \end {align*}
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Mathematica [A] time = 0.04, size = 48, normalized size = 0.81 \[ \frac {4 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )+\sqrt {2} C\left (\sqrt {2} b x\right )-2 b x}{4 \pi b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right ), x\right ) \]
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 \[ \int x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 52, normalized size = 0.88 \[ \frac {\frac {\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b \pi }-\frac {\frac {b x}{2}-\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{4}}{b \pi }}{b} \]
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 \[ \int x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
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 \[ \int x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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