Optimal. Leaf size=8 \[ \cosh (x)-\tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.0075202, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3770, 2638} \[ \cosh (x)-\tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 3770
Rule 2638
Rubi steps
\begin{align*} \int (\text{csch}(x)+\sinh (x)) \, dx &=\int \text{csch}(x) \, dx+\int \sinh (x) \, dx\\ &=-\tanh ^{-1}(\cosh (x))+\cosh (x)\\ \end{align*}
Mathematica [A] time = 0.0035631, size = 10, normalized size = 1.25 \[ \cosh (x)+\log \left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 9, normalized size = 1.1 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) +\cosh \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18091, size = 11, normalized size = 1.38 \begin{align*} \cosh \left (x\right ) + \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 = 1.6715, size = 236, normalized size = 29.5 \begin{align*} \frac{\cosh \left (x\right )^{2} - 2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \,{\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 \left (\sinh{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13579, size = 32, normalized size = 4. \begin{align*} \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} - \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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