Optimal. Leaf size=7 \[ -\frac{1}{2} \tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.0150622, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4287, 3770} \[ -\frac{1}{2} \tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 4287
Rule 3770
Rubi steps
\begin{align*} \int \cosh (x) \text{csch}(2 x) \, dx &=\frac{1}{2} \int \text{csch}(x) \, dx\\ &=-\frac{1}{2} \tanh ^{-1}(\cosh (x))\\ \end{align*}
Mathematica [A] time = 0.0022791, size = 11, normalized size = 1.57 \[ \frac{1}{2} \log \left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 6, normalized size = 0.9 \begin{align*} -{\it Artanh} \left ({{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.07637, size = 26, normalized size = 3.71 \begin{align*} -\frac{1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98568, size = 89, normalized size = 12.71 \begin{align*} -\frac{1}{2} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac{1}{2} \, \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 \cosh{\left (x \right )} \operatorname{csch}{\left (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1409, size = 22, normalized size = 3.14 \begin{align*} -\frac{1}{2} \, \log \left (e^{x} + 1\right ) + \frac{1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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