Optimal. Leaf size=48 \[ \frac {C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {S\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6453, 3351} \[ \frac {\text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {S\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^2} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 6453
Rubi steps
\begin {align*} \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx &=\frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {\int \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac {S\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2 \pi }+\frac {C(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 = 0.03, size = 44, normalized size = 0.92 \[ -\frac {\sqrt {2} S\left (\sqrt {2} b x\right )-4 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right ), x\right ) \]
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 \[ \int x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 45, normalized size = 0.94 \[ \frac {\frac {\FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi b}-\frac {\mathrm {S}\left (b x \sqrt {2}\right ) \sqrt {2}}{4 b \pi }}{b} \]
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 \[ \int x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right )\,{d x} \]
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 \[ \int x\,\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
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 \[ \int x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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