Optimal. Leaf size=6 \[ \text{sech}(x)+\tan ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.035025, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3501, 3770} \[ \text{sech}(x)+\tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3501
Rule 3770
Rubi steps
\begin{align*} \int \frac{\text{sech}^3(x)}{1+\tanh (x)} \, dx &=\text{sech}(x)+\int \text{sech}(x) \, dx\\ &=\tan ^{-1}(\sinh (x))+\text{sech}(x)\\ \end{align*}
Mathematica [A] time = 0.0245494, size = 12, normalized size = 2. \[ \text{sech}(x)+2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 21, normalized size = 3.5 \begin{align*} 2\, \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{-1}+2\,\arctan \left ( \tanh \left ( x/2 \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.58321, size = 30, normalized size = 5. \begin{align*} \frac{2 \, e^{\left (-x\right )}}{e^{\left (-2 \, x\right )} + 1} - 2 \, \arctan \left (e^{\left (-x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.08175, size = 198, normalized size = 33. \begin{align*} \frac{2 \,{\left ({\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + \cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1} \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 \frac{\operatorname{sech}^{3}{\left (x \right )}}{\tanh{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22233, size = 24, normalized size = 4. \begin{align*} \frac{2 \, e^{x}}{e^{\left (2 \, x\right )} + 1} + 2 \, \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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