Integrand size = 2, antiderivative size = 3 \[ \int \coth (x) \, dx=\log (\sinh (x)) \]
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Time = 0.00 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3556} \[ \int \coth (x) \, dx=\log (\sinh (x)) \]
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Rule 3556
Rubi steps \begin{align*} \text {integral}& = \log (\sinh (x)) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(7\) vs. \(2(3)=6\).
Time = 0.00 (sec) , antiderivative size = 7, normalized size of antiderivative = 2.33 \[ \int \coth (x) \, dx=\log (\cosh (x))+\log (\tanh (x)) \]
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Time = 0.06 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.33
method | result | size |
lookup | \(\ln \left (\sinh \left (x \right )\right )\) | \(4\) |
derivativedivides | \(\ln \left (\sinh \left (x \right )\right )\) | \(4\) |
default | \(\ln \left (\sinh \left (x \right )\right )\) | \(4\) |
risch | \(-x +\ln \left ({\mathrm e}^{2 x}-1\right )\) | \(12\) |
parallelrisch | \(\ln \left (\tanh \left (x \right )\right )-\ln \left (1-\tanh \left (x \right )\right )-x\) | \(17\) |
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Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (3) = 6\).
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 6.00 \[ \int \coth (x) \, dx=-x + \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (3) = 6\).
Time = 0.14 (sec) , antiderivative size = 12, normalized size of antiderivative = 4.00 \[ \int \coth (x) \, dx=x - \log {\left (\tanh {\left (x \right )} + 1 \right )} + \log {\left (\tanh {\left (x \right )} \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \coth (x) \, dx=\log \left (\sinh \left (x\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (3) = 6\).
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 4.00 \[ \int \coth (x) \, dx=-x + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \coth (x) \, dx=\ln \left (\mathrm {sinh}\left (x\right )\right ) \]
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