Optimal. Leaf size=5 \[ -\tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.00, antiderivative size = 5, 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 = {3770} \[ -\tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin {align*} \int \text {csch}(x) \, dx &=-\tanh ^{-1}(\cosh (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 7, normalized size = 1.40 \[ \log \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \text {csch}(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.34, size = 17, normalized size = 3.40 \[ -\log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.59, size = 14, normalized size = 2.80 \[ -\log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 6, normalized size = 1.20
method | result | size |
lookup | \(\ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(6\) |
default | \(\ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(6\) |
risch | \(\ln \left (-1+{\mathrm e}^{x}\right )-\ln \left (1+{\mathrm e}^{x}\right )\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 5, normalized size = 1.00 \[ \log \left (\tanh \left (\frac {1}{2} \, x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.01, size = 5, normalized size = 1.00 \[ \ln \left (\mathrm {tanh}\left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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