Optimal. Leaf size=5 \[ -\tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.0036311, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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*}
Mathematica [A] time = 0.0041317, size = 7, normalized size = 1.4 \[ \log \left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 6, normalized size = 1.2 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.936344, size = 7, normalized size = 1.4 \begin{align*} \log \left (\tanh \left (\frac{1}{2} \, x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.12289, size = 78, normalized size = 15.6 \begin{align*} -\log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13312, size = 19, normalized size = 3.8 \begin{align*} -\log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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