Optimal. Leaf size=10 \[ \cosh (x)-\frac{1}{2} \tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.0264988, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {12, 388, 206} \[ \cosh (x)-\frac{1}{2} \tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 12
Rule 388
Rule 206
Rubi steps
\begin{align*} \int \cosh (x) \coth (2 x) \, dx &=-\operatorname{Subst}\left (\int \frac{-1+2 x^2}{2 \left (1-x^2\right )} \, dx,x,\cosh (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+2 x^2}{1-x^2} \, dx,x,\cosh (x)\right )\right )\\ &=\cosh (x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{1}{2} \tanh ^{-1}(\cosh (x))+\cosh (x)\\ \end{align*}
Mathematica [A] time = 0.0144016, size = 14, normalized size = 1.4 \[ \cosh (x)+\frac{1}{2} \log \left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 9, normalized size = 0.9 \begin{align*} \cosh \left ( x \right ) -{\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 = 1.11396, size = 39, normalized size = 3.9 \begin{align*} \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} - \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 = 2.20459, size = 231, normalized size = 23.1 \begin{align*} \frac{\cosh \left (x\right )^{2} -{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) +{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\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 )} \coth{\left (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10755, size = 35, normalized size = 3.5 \begin{align*} \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} - \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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