Optimal. Leaf size=18 \[ -\frac {1}{4} x \cos (4 x)+\frac {1}{16} \sin (4 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3377, 2717}
\begin {gather*} \frac {1}{16} \sin (4 x)-\frac {1}{4} x \cos (4 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3377
Rubi steps
\begin {align*} \int x \sin (4 x) \, dx &=-\frac {1}{4} x \cos (4 x)+\frac {1}{4} \int \cos (4 x) \, dx\\ &=-\frac {1}{4} x \cos (4 x)+\frac {1}{16} \sin (4 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{4} x \cos (4 x)+\frac {1}{16} \sin (4 x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.86, size = 14, normalized size = 0.78 \begin {gather*} -\frac {x \text {Cos}\left [4 x\right ]}{4}+\frac {\text {Sin}\left [4 x\right ]}{16} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 15, normalized size = 0.83
method | result | size |
derivativedivides | \(-\frac {x \cos \left (4 x \right )}{4}+\frac {\sin \left (4 x \right )}{16}\) | \(15\) |
default | \(-\frac {x \cos \left (4 x \right )}{4}+\frac {\sin \left (4 x \right )}{16}\) | \(15\) |
risch | \(-\frac {x \cos \left (4 x \right )}{4}+\frac {\sin \left (4 x \right )}{16}\) | \(15\) |
meijerg | \(\frac {\sqrt {\pi }\, \left (-\frac {2 x \cos \left (4 x \right )}{\sqrt {\pi }}+\frac {\sin \left (4 x \right )}{2 \sqrt {\pi }}\right )}{8}\) | \(26\) |
norman | \(\frac {-\frac {x}{4}+\frac {x \left (\tan ^{2}\left (2 x \right )\right )}{4}+\frac {\tan \left (2 x \right )}{8}}{1+\tan ^{2}\left (2 x \right )}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 14, normalized size = 0.78 \begin {gather*} -\frac {1}{4} \, x \cos \left (4 \, x\right ) + \frac {1}{16} \, \sin \left (4 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 14, normalized size = 0.78 \begin {gather*} -\frac {1}{4} \, x \cos \left (4 \, x\right ) + \frac {1}{16} \, \sin \left (4 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 14, normalized size = 0.78 \begin {gather*} - \frac {x \cos {\left (4 x \right )}}{4} + \frac {\sin {\left (4 x \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -\frac {4}{16} x \cos \left (4 x\right )+\frac {\sin \left (4 x\right )}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 14, normalized size = 0.78 \begin {gather*} \frac {\sin \left (4\,x\right )}{16}-\frac {x\,\cos \left (4\,x\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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