Integrand size = 4, antiderivative size = 9 \[ \int \tanh (2 x) \, dx=\frac {1}{2} \log (\cosh (2 x)) \]
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Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3556} \[ \int \tanh (2 x) \, dx=\frac {1}{2} \log (\cosh (2 x)) \]
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Rule 3556
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \log (\cosh (2 x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \tanh (2 x) \, dx=\frac {1}{2} \log (\cosh (2 x)) \]
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Time = 0.06 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\ln \left (\cosh \left (2 x \right )\right )}{2}\) | \(8\) |
default | \(\frac {\ln \left (\cosh \left (2 x \right )\right )}{2}\) | \(8\) |
risch | \(-x +\frac {\ln \left ({\mathrm e}^{4 x}+1\right )}{2}\) | \(14\) |
parallelrisch | \(-x +\ln \left (\frac {1}{\sqrt {1-\tanh \left (2 x \right )}}\right )\) | \(16\) |
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Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (7) = 14\).
Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.89 \[ \int \tanh (2 x) \, dx=-x + \frac {1}{2} \, \log \left (\frac {2 \, \cosh \left (2 \, x\right )}{\cosh \left (2 \, x\right ) - \sinh \left (2 \, x\right )}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \tanh (2 x) \, dx=\frac {\log {\left (\cosh {\left (2 x \right )} \right )}}{2} \]
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none
Time = 0.19 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \tanh (2 x) \, dx=\frac {1}{2} \, \log \left (\cosh \left (2 \, x\right )\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.44 \[ \int \tanh (2 x) \, dx=-x + \frac {1}{2} \, \log \left (e^{\left (4 \, x\right )} + 1\right ) \]
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Time = 0.19 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \tanh (2 x) \, dx=\frac {\ln \left (\mathrm {cosh}\left (2\,x\right )\right )}{2} \]
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