Integrand size = 6, antiderivative size = 11 \[ \int \tanh (a+b x) \, dx=\frac {\log (\cosh (a+b x))}{b} \]
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Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3556} \[ \int \tanh (a+b x) \, dx=\frac {\log (\cosh (a+b x))}{b} \]
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Rule 3556
Rubi steps \begin{align*} \text {integral}& = \frac {\log (\cosh (a+b x))}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \tanh (a+b x) \, dx=\frac {\log (\cosh (a+b x))}{b} \]
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Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\ln \left (\cosh \left (b x +a \right )\right )}{b}\) | \(12\) |
default | \(\frac {\ln \left (\cosh \left (b x +a \right )\right )}{b}\) | \(12\) |
parallelrisch | \(-\frac {b x +\ln \left (1-\tanh \left (b x +a \right )\right )}{b}\) | \(21\) |
risch | \(-x -\frac {2 a}{b}+\frac {\ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b}\) | \(27\) |
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (11) = 22\).
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 3.36 \[ \int \tanh (a+b x) \, dx=-\frac {b x - \log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (8) = 16\).
Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55 \[ \int \tanh (a+b x) \, dx=\begin {cases} x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} & \text {for}\: b \neq 0 \\x \tanh {\left (a \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \tanh (a+b x) \, dx=\frac {\log \left (\cosh \left (b x + a\right )\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (11) = 22\).
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 2.18 \[ \int \tanh (a+b x) \, dx=-\frac {b x + a - \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} \]
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Time = 1.69 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.45 \[ \int \tanh (a+b x) \, dx=x-\frac {\ln \left (\mathrm {tanh}\left (a+b\,x\right )+1\right )}{b} \]
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