Integrand size = 7, antiderivative size = 29 \[ \int \cos (\log (6+3 x)) \, dx=\frac {1}{2} (2+x) \cos (\log (3 (2+x)))+\frac {1}{2} (2+x) \sin (\log (3 (2+x))) \]
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Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4564} \[ \int \cos (\log (6+3 x)) \, dx=\frac {1}{2} (x+2) \sin (\log (3 (x+2)))+\frac {1}{2} (x+2) \cos (\log (3 (x+2))) \]
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Rule 4564
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}(\int \cos (\log (x)) \, dx,x,6+3 x) \\ & = \frac {1}{2} (2+x) \cos (\log (3 (2+x)))+\frac {1}{2} (2+x) \sin (\log (3 (2+x))) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \cos (\log (6+3 x)) \, dx=\frac {1}{2} (2+x) (\cos (\log (3 (2+x)))+\sin (\log (3 (2+x)))) \]
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Time = 0.81 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {\cos \left (\ln \left (6+3 x \right )\right ) \left (6+3 x \right )}{6}+\frac {\left (6+3 x \right ) \sin \left (\ln \left (6+3 x \right )\right )}{6}\) | \(30\) |
risch | \(\left (\frac {1}{4}-\frac {i}{4}\right ) \left (2+x \right ) \left (6+3 x \right )^{i}+\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2+x \right ) \left (6+3 x \right )^{-i}\) | \(34\) |
parallelrisch | \(\frac {2 x \tan \left (\ln \left (\sqrt {6+3 x}\right )\right )-\tan \left (\ln \left (\sqrt {6+3 x}\right )\right )^{2} x +2 \tan \left (\ln \left (\sqrt {6+3 x}\right )\right )^{2}+4 \tan \left (\ln \left (\sqrt {6+3 x}\right )\right )+x +6}{2 \tan \left (\ln \left (\sqrt {6+3 x}\right )\right )^{2}+2}\) | \(72\) |
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Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \cos (\log (6+3 x)) \, dx=\frac {1}{2} \, {\left (x + 2\right )} \cos \left (\log \left (3 \, x + 6\right )\right ) + \frac {1}{2} \, {\left (x + 2\right )} \sin \left (\log \left (3 \, x + 6\right )\right ) \]
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\[ \int \cos (\log (6+3 x)) \, dx=\int \cos {\left (\log {\left (3 x + 6 \right )} \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \cos (\log (6+3 x)) \, dx=\frac {1}{2} \, {\left (x + 2\right )} {\left (\cos \left (\log \left (3 \, x + 6\right )\right ) + \sin \left (\log \left (3 \, x + 6\right )\right )\right )} \]
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Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \cos (\log (6+3 x)) \, dx=\frac {1}{2} \, {\left (x + 2\right )} \cos \left (\log \left (3 \, x + 6\right )\right ) + \frac {1}{2} \, {\left (x + 2\right )} \sin \left (\log \left (3 \, x + 6\right )\right ) \]
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Time = 25.67 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \cos (\log (6+3 x)) \, dx=\frac {\sqrt {2}\,\sin \left (\frac {\pi }{4}+\ln \left (3\,x+6\right )\right )\,\left (3\,x+6\right )}{6} \]
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