Integrand size = 3, antiderivative size = 17 \[ \int \cos (\log (x)) \, dx=\frac {1}{2} x \cos (\log (x))+\frac {1}{2} x \sin (\log (x)) \]
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Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4564} \[ \int \cos (\log (x)) \, dx=\frac {1}{2} x \sin (\log (x))+\frac {1}{2} x \cos (\log (x)) \]
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Rule 4564
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \cos (\log (x))+\frac {1}{2} x \sin (\log (x)) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \cos (\log (x)) \, dx=\frac {1}{2} x \cos (\log (x))+\frac {1}{2} x \sin (\log (x)) \]
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Time = 0.08 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65
method | result | size |
parallelrisch | \(\frac {x \left (\cos \left (\ln \left (x \right )\right )+\sin \left (\ln \left (x \right )\right )\right )}{2}\) | \(11\) |
lookup | \(\frac {x \cos \left (\ln \left (x \right )\right )}{2}+\frac {x \sin \left (\ln \left (x \right )\right )}{2}\) | \(14\) |
default | \(\frac {x \cos \left (\ln \left (x \right )\right )}{2}+\frac {x \sin \left (\ln \left (x \right )\right )}{2}\) | \(14\) |
risch | \(\left (\frac {1}{4}-\frac {i}{4}\right ) x \,x^{i}+\left (\frac {1}{4}+\frac {i}{4}\right ) x \,x^{-i}\) | \(22\) |
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none
Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (\log (x)) \, dx=\frac {1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac {1}{2} \, x \sin \left (\log \left (x\right )\right ) \]
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Time = 0.13 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \cos (\log (x)) \, dx=\frac {x \sin {\left (\log {\left (x \right )} \right )}}{2} + \frac {x \cos {\left (\log {\left (x \right )} \right )}}{2} \]
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none
Time = 0.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \cos (\log (x)) \, dx=\frac {1}{2} \, x {\left (\cos \left (\log \left (x\right )\right ) + \sin \left (\log \left (x\right )\right )\right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (\log (x)) \, dx=\frac {1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac {1}{2} \, x \sin \left (\log \left (x\right )\right ) \]
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Time = 0.16 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (\log (x)) \, dx=\frac {\sqrt {2}\,x\,\sin \left (\frac {\pi }{4}+\ln \left (x\right )\right )}{2} \]
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