Optimal. Leaf size=3 \[ \log (\sinh (x)) \]
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Rubi [A] time = 0.00, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3475} \[ \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rubi steps
\begin {align*} \int \coth (x) \, dx &=\log (\sinh (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 3, normalized size = 1.00 \[ \log (\sinh (x)) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth (x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.12, size = 18, normalized size = 6.00 \[ -x + \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 12, normalized size = 4.00 \[ -x + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 4, normalized size = 1.33
method | result | size |
lookup | \(\ln \left (\sinh \relax (x )\right )\) | \(4\) |
default | \(\ln \left (\sinh \relax (x )\right )\) | \(4\) |
risch | \(-x +\ln \left (-1+{\mathrm e}^{2 x}\right )\) | \(12\) |
derivativedivides | \(-\frac {\ln \left (\coth \relax (x )-1\right )}{2}-\frac {\ln \left (\coth \relax (x )+1\right )}{2}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 3, normalized size = 1.00 \[ \log \left (\sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 3, normalized size = 1.00 \[ \ln \left (\mathrm {sinh}\relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.31, size = 12, normalized size = 4.00 \[ x - \log {\left (\tanh {\relax (x )} + 1 \right )} + \log {\left (\tanh {\relax (x )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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