Optimal. Leaf size=48 \[ -\frac {S\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2 \pi }+\frac {\text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi } \]
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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 = {6588, 3432}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 6588
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 = 0.02, size = 44, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {2} S\left (\sqrt {2} b x\right )-4 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^2 \pi } \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.61, size = 45, normalized size = 0.94
method | result | size |
default | \(\frac {\frac {\FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi b}-\frac {\mathrm {S}\left (b x \sqrt {2}\right ) \sqrt {2}}{4 b \pi }}{b}\) | \(45\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 47, normalized size = 0.98 \begin {gather*} \frac {4 \, b \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{4 \, \pi b^{3}} \end {gather*}
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 \begin {gather*} \int x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x\,\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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