Optimal. Leaf size=3 \[ \tan ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.0032907, antiderivative size = 3, 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} \[ \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int \text{sech}(x) \, dx &=\tan ^{-1}(\sinh (x))\\ \end{align*}
Mathematica [B] time = 0.0039811, size = 9, normalized size = 3. \[ 2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 4, normalized size = 1.3 \begin{align*} \arctan \left ( \sinh \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.927777, size = 4, normalized size = 1.33 \begin{align*} \arctan \left (\sinh \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.08164, size = 39, normalized size = 13. \begin{align*} 2 \, \arctan \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 \operatorname{sech}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09022, size = 7, normalized size = 2.33 \begin{align*} 2 \, \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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