Optimal. Leaf size=59 \[ -\frac {x}{2 b \pi }+\frac {\text {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2 \pi }+\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi } \]
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Rubi [A]
time = 0.02, 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 = {6595, 3438,
3433} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3433
Rule 3438
Rule 6595
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 = 0.02, size = 48, normalized size = 0.81 \begin {gather*} \frac {-2 b x+\sqrt {2} \text {FresnelC}\left (\sqrt {2} b x\right )+4 S(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 = 0.81, size = 52, normalized size = 0.88
method | result | size |
default | \(\frac {\frac {\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b \pi }-\frac {\frac {b x}{2}-\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{4}}{b \pi }}{b}\) | \(52\) |
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 = 0.37, size = 53, normalized size = 0.90 \begin {gather*} -\frac {2 \, b^{2} x - 4 \, b \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - \sqrt {2} \sqrt {b^{2}} \operatorname {C}\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 )} S\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 {FresnelS}\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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